^<^  /^-/^ 


Compliments  of 


ELI  W.   BLAKE. 


ORIGINAL  SOLUTIONS 


SEVERAL    PROBLEMS 


AERODYNAMICS. 


By  ELI  W.  BLAKE. 


NEW    HAVEN: 
TUTTLE,    MOREHOUSE    &    TAYLOR,    PRINTERS. 

1882. 


.0 


^\^ 


PREFACE. 


The  employment  of  the  laws  and  principles  of  dynamics  in  dis- 
cussing the  movements  of  the  parts  of  an  aerial  medium  caused 
by  the  action  of  a  local  force  impressed  upon  them,  or  by  their 
own  inherent  elastic  force  causing  them  to  flow  towards  a  total 
or  partial  vacuum,  has  heretofore  been  thought  by  mathemati- 
cians to  be  attended  with  great  difficulties  ;  so  great  indeed  that 
few  attempts  were  made  to  surmount  them,  and  none  that  were 
attended  with  full  and  complete  success.  Hence  many  problems 
in  aerodynamics,  that  were  of  great  interest  to  science  and  the 
arts,  remained  unsolved.  •  Some  years  since  I  became  deeply 
interested  in  an  attempt  to  solve  one  of  these  problems,  and 
after  much  thought  succeeded  in  devising  a  method  whereby  a 
satisfactory  solution  was  accomplished.  Pert^eiving  that  the 
same  method,  with  suitable  modifications,  was  applicable  to  the 
solution  of  other  problems,  I  gave  my  thoughts  to  others  from 
time  to  time  as  I  found  leisure  amid  the  active  pursuits  of  life  ; 
and  the  results  of  these  investigations,  as  they  were  respectively 
reached,  were  published  in  the  then  current  numbers  of  the 
American  Journal  of  Science. 

Believing  that  the  methods  employed  in  those  investigations 
might  be  employed  with  success  in  solving  other  important 
problems,  and  desiring  to  bring  this  field  of  research  to  the 
attention  of  physicists,  1  recently  prepared  another  paper  per- 
taining to  this  branch  of  physics  and  offered  it  for  publication 
in  the  Journal  of  Science  ;  but  the  editors,  finding  in  it  positions 
and  conclusions  which  conflicted  with  their  pre-conceived 
opinions ;  and  not  being  disposed  to  publish  that  which  they 
Avere  not  ready  to  endorse,  declined  the  article. 

Under  these  circumstances  it  was  decided  to  print  the  paper, 
prefixing  to  it  the  articles  that  had  been  published  in  the 
Journal  of  Science  pertaining  to  aerodynamics,  arranged  in  the 
order  of  their  respective  dates ;  and  in  this  collective,  but  some- 
what disjointed  form,  to  present  them  as  a  contribution  toward 
a  more  full  development  of  this  interesting  and  important 
branch  of  Physics. 


ARTICLE  I. 


A   Theoretical  Determination  of  the  Law  which  governs 
the  Flow  of  Elastic  Fluids  through  Orifices* 

The  subject  announced  at  the  head  of  this  article,  is  not  only 
interesting  considered  simply  as  a  subject  of  scientific  inquiry, 
but  it  is  also  a  matter  of  practical  importance  in  its  relations  to 
several  branches  of  mechanism.  Among  these  may  be  instanced, 
as  perhaps  first  in  importance,  the  bearing  of  the  subject  upon 
the  construction  of  the  steam  engine.  The  size  of  the  pipes  and 
valves  which  conduct  the  steam  to  and  from  the  working  cylin- 
der, should  be  properly  adjusted  to  the  size  and  velocity  of  the 
piston.  In  general,  the  larger  these  pipes  and  valves  the  better, 
so  far  as  respects  the  power  of  the  engine.  But  there  are  incon- 
veniences attendant  on  making  them  large;  and  in  order  to 
make  a  due  compromise  between  the  inconvenience  that  may  be 
incurred  on  the  one  hand  and  the  amount  of  power  that  may  be 
sacrificed  on  the  other,  it  becomes  necessary  to  understand  cor- 
rectly the  law  which  governs  the  flow  of  elastic  fluids  through 
orifices.  Treatises  on  the  dynamics  of  fluids  have  not  omitted  to 
give  rules  for  the  determination  of  such  questions;  but  it  will  be 
seen  in  the  course  of  this  article  that  those  rules  are  very  defec- 
tive. 

If  the  velocity  with  which  a  fluid  flows  through  an  orifice  from 
one  vessel  into  another  be  represented  by  Y,  the  density  under 
which  it  passes  the  orifice  by  D,  and  the  area  of  the  orifice  by  S, 
then  the  product  VDS  is  the  measure  of  the  quantity  of  fluid 
discharged  in  a  given  time.  It  is  an  established  law  in  the 
dynamics  of  fluids,  that  the  velocity  of  the  flow  is  directly  as  the 

*  Published  A.  D.  1848,  in  the  American  Jotirnal  of  Science,  second  series,  vol. 
V,  page  78. 

2 


square  root  of  the  pressure  and  inversely  as  the  square  root  of 
the  density.  If,  then,  the  eflScient  pressure  which  produces  this 
flow  be  represented  by  P,  the  general  law  expressed  by  symbols 
will  be, 

Dsyp 

The  above  expression  is  in  accordance  with  the  received  theory, 
and  properly  understood  it  is  correct  and  applicable  to  all  fluids, 
elastic  as  well  as  inelastic.  But  it  must  be  observed  that  D  in 
this  expression  must  in  all  cases  represent  the  density  utider 
which  the  fluid  passes  the  orifice. 

In  all  the  treatises  on  the  dynamics  of  fluids  that  I  have  exam- 
ined, the  quantity  D  in  the  foregoing  expression  represents  the 
density  of  the  fluid  in  the  discharging  vessel:  it  being  assumed 
that  the  fluid  passes  the  orifice  without  change  of  density.  This 
assumption  is  correct  so  far  as  respects  inelastic  fluids,  but  as 
respects  elastic  fluids  it  is  far  otherwise.  A  particle  cannot  even 
begin  to  approach  the  orifice  without  a  change  of  density.  Sur- 
rounded by  other  particles,  it  will  not  begin  to  move  until  the 
pressure  before  it  becomes  less  than  the  pressure  behind  it.  If 
the  pressure  before  it  is  less  than  the  pressure  behind  it,  then  the 
density  there  is  less  also,  and  consequently  the  density  of  the 
particle  itself  is  diminished,  for  that  must  be  intermediate  be- 
tween the  density  before  and  behind  it;  and  as  it  cannot  begin 
to  move  without  a  change  of  density,  so  for  the  same  reasons  its 
motion  cannot  be  accelerated  without  a  further  change  of  density. 
Thus  for  every  increment  of  its  velocity  in  its  approach  to  the 
orifice,  there  must  be  a  corresponding  decrement  of  its  density. 
Hence  it  is  evident  that  the  fluid  passes  the  orifice  under  a  den- 
sity less  than  the  density  in  the  discharging  vessel. 

Again,  there  is  an  error  in  the  received  theory,  in  considering 
the  efticient  pressure  which  causes  the  discharge  (represented  by 
P  in  the  above  expression),  as  equal  in  all  cases  to  the  difference 
of  pressure  in  the  two  vessels.  The  true  amount  which  is  to  be 
deducted  from  the  pressure  in  the  discharging  vessel,  in  order  to 
find  the  eflicient  pressure  that  produces  the  discharge,  is  the 
elastic  force  that  is  due  to  the  density  which  the  fluid  has  in  its 
passage  through  the  orifice;  for  it  is  obvious  that  that  alone 
reacts  against  the  pressure  in  the  discharging  vessel.  From  this 
consideration  also  we  may  arrive  at  the  same  conclusion  as  was 
deduced  in  the  last  paragraph,  viz:  that  the  fluid  must  pass  the 


orifice  with  a  diminished  density;  for  otherwise  the  elastic  force 
of  the  fluid  in  the  orifice  would  he  a  perfect  counterpoise  to  the 
pressure,  and  there  could  be  no  flow. 

From  what  precedes  it  will  be  apparent  that  in  the  application 
of  the  general  expression, 


VDS 


-v/D 


to  the  case  of  elastic  fluids,  the  density  of  the  fluid  in  the  orifice, 
represented  by  D,  is  an  unknown  quantity,  having  a  value  some- 
where intermediate  between  cipher  and  the  density  of  the  fluid 
in  the  discharging  vessel;  also  that  the  efficient  pressure  which 
produces  the  discharge,  represented  by  P,  is  an  unknown  quantity 
whose  value  is  dependent  on  that  of  D.  We  will  now  proceed  to 
elicit  a  general  rule  for  the  determination  of  the  value  of  D  and 
P  in  every  case  that  can  occur. 

In  the  annexed  figure  let  z/  be  the  dis- 
charging vessel  containing  fluid  whose 
density  is  Zf,  and  let  d  be  the  receiving 
vessel,  which  for  the  present  we  will 
consider  a  vacuum.  Let  the  smallest 
place  in  the  passage  leading  from  one  to 
the  other  be  the  orifice,  and  let  its  area 
be  S,  and  let  D  represent  the  unknown  density  with  which  the 
fluid  passes  the  orifice.  Since  the  pressure  is  as  the  density,  the 
density  may  be  employed  to  express  the  pressure.  Then  it  fol- 
lows from  the  preceding  observations  that  the  efficient  pressure 
which  produces  the  discharge  is  z/  —  D.  Since  the  velocity  will 
be  directly  as  the  square  root  of  the  pressure  and  inversely  as 
the  square  root  of  the  density,  we  have, 

x/(z.r-D) 


V 


VO 


.Multiplying  this  expression  by  D  and  reducing,  we  have, 

VD  X -v/(^D-D^)- 

Now  if  we  conceive  several  sections  to  be  made  across  the 
passage  at  different  points  on  each  side  of  the  orifice,  and  if  the 
areas  of  these  sections  are  respectively  S',  S",  etc.,  the  velocities 
of  the  fluid  in  them  V,  V  ',  etc.  and  the  densities  D',  D ",  etc. 
V'D'S',  V"D"S",  etc.  are  the  measures  of  the  quantities  of  fluid 
that  pass  through  these  sections  respectively  in  a  given  time. 


8 

l>ut  when  the  current  is  established,  the  same  quantity  flows 
through  each  in  a  given  time.  Therefore  VD'S'^Y'D'S' 
=VDS.  Now  VDS  being  a  constant  quantity,  if  each  of  the 
factors  vary,  VD  will  be  a  maximum  when  S  is  a  minimum. 
But  S  is  a  minimum  at  the  orifice;  and  therefore  VD  is  a  maxi- 
mum at  the  orifice.  But  we  have  before  found  YD  x  v'C^D  — D^); 
and  therefore  when  VD  is  a  maximum  \/(^D— D^)  must  likewise 

be  a  maximum.     Now,  when  /^^(z^D— D^)  is  a  maximum  D  =  — . 

Hence,  when  the  discharge  is  into  a  vacuum,  the  density  of  the 
fluid  at  the  orifice  is  equal  to  half  the  density  in  the  discharging 
vessel. 

For  convenience  in  the  illustration,  we  have  made  the  passage 
from  one  vessel  to  the  other  in  the  figure,  divergent  each  way 
from  the  orifice;  but  our  reasoning  would  obviously  be  equally 
applicable  if  the  orifice  opened  directly  from  one  vessel  into  the 
other,  without  the  intervention  of  the  divergent  tubes. 

Let  us  now  inquire,  what  will  be  the  value  of  D  when  the 
receiving  vessel  contains  fluid  of  any  density  less  than  that  in 
the  discharging  vessel. 

Let  the  density  in  the  receiving  vessel  be  d.  Theji  J  is  a  limit 
beyond  which  the  fluid  cannot  expand,  either  before  or  after  it 
passes  the  orifice;  so  that  D  can  never  be  less  that  d.  As  in  the 
preceding  case  /s/{AT>—J}^)  was  the  maximum  for  all  the  values 
that  can  be  assigned  from  cipher  to  z/,  so  in  this  case,  and  for 
the  same  reasons,  \/(zJD— D^)  must  be  the  maximum  for  all  the 
values  of  D  that  can  be  assigned  between  d  and  Z'.     But  we 

found  in  the  other  case  that  the  maximum  occurs  when  D  =  — . 

2 

If,  then,  this  value  of  D  is  assignable  between  d  and  J,  the 

maximum  must  in  this  case  also  occur  when  D  =  — .     But  this 

2 

value  of  D  will  always  be  assignable  between  d  and  J,  if  d  be 

not  greater  than  — .     Therefore  if  d  be  any  quantity  not  greater 

A  A  •.  . 

than  — ,  D  will  be  equal  to  — .     In  other  words,  if  the  density  in 

the  receiving  vessel  be  not  greater  than  half  the  density  in  the 
discharging  vessel,  the  density  in  the  orifice  will  be  equal  to  half 
the  density  in  the  discharging  vessel. 

Again,  from  the  nature  of  maxima  and  minima,  it  is  obvious 


9 

that  \^[/]l>—D^)  will  be  a  maximum  when,  of  all  the  values  that 
are  assignable  to  D,  that  value  is  assigned  which  differs  least 

from  — .     Hence,  if  d  exceed  — ,  so  that  D  must  have  a  value 

greater  than  — ,  then  \/(z?D— D^)  wall  be  a  maximum  when  D  has 

the  smallest  value  that  is  assignable  to  it.  Now  the  smallest 
value  that  is  assignable  to  it  in  this  case  is  D  =  d.  Hence,  if  the 
density  in  the  receiving  vessel  exceed  half  the  density  in  the 
discharging  vessel,  the  density  under  which  the  fluid  passes  the 
orifice  is  equal  to  the  density  in  the  receiving  vessel. 

Thus  we  have  found  for  the  value  of  D,  D  =  —  if  c?  is  not 

greater  than  — ;  if  otherwise,  D  =  d.     And  for  the  value  of  P 

(since  P  =  z/-D)  we  have  P  =  —  if  d  is  not  greater  than  — ;  if 
otherwise,  P  =  z/  —  f7. 

In  applying  the  general  dynamic   law  VDS  x  —  to  the 

case  of  elastic  fluids,  the  values  of  D  and  P  should  therefore  be 
assigned  in  accordance  with  this  rule. 

We  have  already  remarked  that  treatises  on  the  dynamics  of 
fluids,  in  applying  the  above  general  expression  to  elastic  fluids, 
put  D  as  equal  to  J,  and  P  as  equal  to  A—  d  in  all  cases.  This, 
as  will  appear  from  the  above  rule,  makes  D  too  large  in  all 
cases;  and  P  also  too  large  whenever  d  is  less  than  half  z/.  In 
.the  case  of  a  discharge  into  a  vacuum,  it  makes  each  of  these 
quantities  double  what  it  should  be.  In  constructing  a  formula 
to  express  the  velocity  of  the  flow  into  a  vacuum,  these  errors 
balance  each  other;  so  that  in  that  particular  case  the  result  is 
the  same  as  if  the  values  of  these  quantities  were  assigned  in 

2 
accordance  with  our  rule;  for  by  our  rule  V  x        --  =  1,  and  by 


^  2 


the  old  rule  V  x  ~^-^  =  1. 
\/z/ 


2   . 
Again,  since  — -^  is  a  constant  quantity  however  z/  may  vary, 

^2" 


10 

it  follows  from  our  rule  that  the  velocity  of  the  flow  into  a 
vacuum  is  a  constant  quantity,  being  the  same  for  every  density 
in  the  discharging  vessel.  The  same  also  results  from  the  old 
theory. 

But  in  constructing  a  formula  to  express  the  quantity  that  will 
flow  into  a  vacuum  in  a  given  time,  the  results  of  the  two  rules 
will  differ  widely.  For  since  the  rule  gives  the  velocity  of  the 
flow  correctly,  and  at  the  same  time  puts  its  density  at  double 
what  it  should  be,  it  follows  that  the  old  rule  makes  the  quantity 
discharged  in  a  given  time  double  what  it  should  be. 

Hence  it  appears  that  in  a  steam  engine,  the  valves  and  pipes 
which  convey  the  steam  from  the  working  cylinder  to  the  con- 
denser, must  be  of  double  the  size  that,  would  be  assigned  to 
them  by  the  old  rule  in  order  to  discharge  the  contents  of  the 
working  cylinder  in  a  given  time,  without  increased  reaction 
upon  the  piston. 

It  appears  from  the  rule  as  stated  above,  for  finding  the  values 
of  D  and  P,  that  the  quantity  c?,  which  expresses  the  density  of 
the  fluid  in  the  receiving  vessel,  will  not  enter  at  all  into  the 
formula  which  expresses  the  velocity  of  the  flow,  nor  into  that 
which  expresses  the  quantity  discharged  in  a  given  time,  pro- 
vided it  be  equal  to,  or  less  than,  half  the  density  in  the  discharg- 
ing vessel.  Hence  it  follows  that  the  fluid  in  the  receiving  vessel, 
if  its  density  does  not  exceed  half  the  density  in  the  discharging 
vessel,  will  have  no  effect  whatever  upon  the  flow.  Consequently 
air  or  steam  will  rush  into  a  vacuum  no  faster  than  into  a  vessel 
containing  flifid  of  half  its  density.  On  the  contrary,  both  the 
velocity  of  the  flow  and  the  quantity  discharged  in  a  given  time 
will  be  the  same  in  both  cases;  and  so,  also,  if  the  density  in  the 
receiving  vessel  is  any  quantity  less  than  half  the  density  in  the 
discharging  vessel,  the  flow  will  be  the  same  in  velocity,  quantity, 
and  density,  as  into  a  vacuum.  Accordingly,  a  vessel  containing 
steam  of  a  density  due  to  a  pressure  of  ten  or  any  other  number 
of  atmospheres,  will  empty  itself  no  faster  into  a  vacuum  than 
into  the  open  air,  until  in  the  progress  of  the  discharge,  the  den- 
sity is  reduced  below  that  due  to  the  pressure  of  two  atmospheres. 
It  will  be  readily  seen  that  these  conclusions  have  an  important 
bearing  upon  the  construction  of  the  steam  engine. 


ARTICLE  II. 


A  Determination  of  the  general  law  of  the  Propagation 
of  Pulses  in  Elastic  Media* 

Sir  Isaac  Newton  in  his  Principia,  Book  II,  prop.  47,  48,  49, 
has  determined  the  velocity  of  a  pulse,  propagated  in  the  atmos- 
phere, whose  intensity  differs  from  cipher  only  by  a  very  small 
quantity;  or  rather,  as  we  shall  see  in  the  course  of  this  article, 
whose  intensity  is  cipher.  This  velocity  he  shows  to  be  that 
which  a  body  would  acquire  by  falling  over  half  the  atmospherical 
subtangent  (or  half  the  height  of  a  homogeneous  atmosphere); 
and  this  he  assumes  would  be  the  velocity  of  sound,  provided  the 
atmosphere  were  perfectly  pure  and  perfectly  elastic.  Lagrange 
and  others  have  since  investigated  the  same  case  by  different 
processes,  but  with  the  same  result,  and  have  concurred  with 
Newton  in  regarding  that  result  as  showing  the  true  theoretical 
velocity  of  sound.  Both  Newton  and  Lagrange  in  their  respect- 
ive solutions  of  the  case  assume,  in  effect,  that  the  velocity  of 
the  pulse  is  irrespective  of  its  intensity.  Newton  at  one  point 
(prop.  48,  case  1),  appears  to  recognize  the  fact  that  an  increased 
intensity  would  make  a  difference,  but  thinks  that  vmless  the 
pulse  were  "  exceedingly  intense,"  the  error  would  not  be  sensible. 
But  Lagrange  says  (Mecanique  Analytique,  Part  II,  sec.  1 2,  art. 
14),  "the  velocity  of  the  pulse  is  constant  and  independent  of 
the  primitive  movement,  which  is  confirmed  by  experience,  as  all 
sounds  strong  or  weak  appear  to  be  propagated  with  the  same 
velocity." 

It  is  proposed  in  this  article  to  show  that  the  velocities  of 
pulses  vary  with  their  intensities,  and  to  determine,  in  general 

*  Published  A.  D.  184P,  in  the  American  Journal  of  Science,  second  series,  vol. 
V,  p.  372. 


12 

form,  the  relation  which  subsists  between  the  velocity  and  inten- 
sity of  pulses.  This  we  shall  do  by  solving  the  general  problem 
of  the  velocity  of  pulses  by  a  new  process,  which  comprehends 
the  intensity  of  the  pulse  as  an  essential  element.  Having  done 
this,  we  shall  see  the  relation  in  whicii  the  case  solved  by  Newton 
stands  to  the  general  law;  and  if  we  mistake  not,  it  will  then  be 
apparent  that  the  velocity  found  in  that  case  is  not  the  velocity 
of  sound,  but  a  lintit  below  which  its  velocity  cannot  fall. 

A  pulse,  considered  as  propagated  through  a  line  of  particles, 
and  considered  with  reference  to  its  physical  condition  at  any 
instant  of  time,  consists  of  a  series  of  contiguous  particles  in 
that  line,  greater  or  less  in  number,  which  are  more  dense  than 
the  particles  before  and  behind  them  on  the  line,  and  which  are 
in  motion  with  some  velocity,  while  the  particles  before  and 
behind  are  at  rest  and  in  their  natural  state  of  density.  This 
series  of  particles,  as  it  advances,  encounters  successively  the 
stationary  particles,  compressing  them  to  the  same  density,  put- 
ting them  in  motion,  and  thus  adds  them  to  the  series.  In  the 
mean  time,  an  equal  number  of  the  posterior  particles  of  the 
series  expand,  resuming  their  natural  density,  and  come  to  rest. 
It  is  obvious  that  if  the  propelling  force  due  to  the  reaction  of 
the  particles  expanding  from  the  }iosterior  extremity  of  the  series, 
is  equal  to  the  retarding  force  of  the  particles  encountered  by 
the  anterior  extremity  (as  must  always  be  the  case  if  the  elasticity 
is  perfect,  and  if  the  action  is  confined  to  the  particles  in  the 
line),  then  the  pulse  will  continue  to  advance  indefinitely,  and 
with  a  uniform  velocity;  a  velocity  however  which,  as  we  shall 
see,  is  not  independent  of  the  degree  of  condensation  to  which 
the  particles  are  brought. 

Fig.  1. 
C  D 

A  JL I Tl 


Let  C  be  a  point  where  the  density  is  a  maximum  in  a  pulse 
which  is  moving  toward  B.  It  is  not  material  to  our  present 
purpose  to  inquire  whether  the  place  of  maximum  density  is  a 
mere  point,  or  whether  it  extends  over  some  finite  space  on  the 
line  AB.  In  either  case,  somewhere  in  advance  of  C  we  shall 
find  particles  in  every  stage  of  density  from  the  natural  to  the 
maximum  state;  and  these  will  be  arranged  in  the  order  of  their 
density,  the  more  dense  being  toward  C  Each  of  these  })articles 
will  be  accelerated  so  long  as  the  density  of  the  jjarticle  behind 


13 

it  is  greater  than  that  of  the  particle  before  it,  and  no  longer- 
Consequently,  each  will  have  its  maximum  velocity  when  it 
reaches  its  maximum  density.  Therefore,  if  C  is  a  point  where 
the  density  is  a  maximum,  it  is  likewise  a  point  where  the  velocity, 
which   the   pulse   gives   successively   to   all   the   particles,  is  a 

maximum. 

Fig.  2. 


Let  mn  be  the  space  which  a  particle  in  its  natural  state  occu- 
pies on  the  line  AB,  fig.  1  and  let  sn  be  that  which  it  occupies  in 
its  most  condensed  state.  Let  D  be  a  point  in  the  line  AB,  in 
advance  of  C,  where  the  particles  are  at  rest,  not  having  yet  felt 
the  influence  of  the  approaching  pulse;  and  let  the  two  imaginary 
points  C  and  D  be  conceived  to  move  with  the  same  velocity  as 
the  pulse.  Then  each  travels  over  the  same  space  and  through 
the  same  number  of  particles  in  a  given  time.  Consequently 
while  C  moves  over  a  space  equal  to  that  occupied  by  a  particle 
in  its  natural  state,  it  only  moves  through  one  particle  in  its  most 
condensed  state;  that  is  to  say,  while  C  moves  over  the  space  mn, 
it  moves,  relatively  to  the  particle  through  which  it  passes,  only 
over  sn.  Consequently  the  particle  itself  moves  in  the  same  time 
over  a  space  equal  to  ms.  Hence  when  mn  represents  the  velocity 
of  the  pulse,  ms  represents  the  final  velocity  which  the  pulse 
gives  to  every  particle  through  which  it  passes. 

Let  H  be  the  atmospherical  subtangent;  or  the  length  of  the 
column  of  particles  of  the  natural  density  whose  weight  is  equal 
to  the  elastic  force  of  a  particle  in  its  natural  state,  and  let  Yi-\-h 
be  the  length  of  a  similar  column  whose  weight  is  equal  to  the 
elastic  force  of  a  particle  at  the  maximum  density.  Since  the 
space  occupied  by  a  particle  is  inversely  as  the  compressing  force, 
we  have, 

mn  :  soi  : :  Yl-\-h  :  H   or  nis-^sn  :sn  ::  H-\-h  :  H  ; 

sn  X  h 

consequently  ms  :  sn  : :  A  :  H.      Whence  H= 

ins 

The  force  which  accelerates  all  the  particles  in  advance  of  C, 
which  have  felt  the  influence  of  the  pulse  but  have  not  yet 
reached  their  maximum  velocity,  is  the  difference  of  the  elastic 
forces  which  correspond  to  the  natural  and  the  maximum  densi- 
ties. This  is  a  constant  force,  and  must  evidently  be  that  force 
which  is  competent  to  give  the  velocity  ms  to  all  the  particles  in 
3 


14 

any  space  in  the  time  in  which  the  pulse  runs  over  that  space. 
Let  us  suppose  the  pulse  runs  over  the  space  h.  Then  the  pulse 
runs  over  h  in  the  time  in  which  the  difference  of  those  elastic 
forces  will  give  to  all  the  particles  in  h  the  velocity  ms.  But  the 
difference  of  the  elastic  forces  is  equal  to  the  weight  of  all  the 
particles  in  h.  Therefore  the  pulse  runs  over  A  in  the  time 
in  which  those  particles  would  in  falling  by  their  own  weight 
acquire  the  velocity  ms. 

The  time  in  which  a  falling  body  acquires  the  velocity  ms  is  to 
the  time  in  which  it  would  acquire  the  velocity  of  the  pulse  or 
mn,  as  ms  to  mn,  and  the  spaces  over  which  the  pulse  would  run 
in  these  times  are  as  the  times  and  therefore  as  ms  to  mii.  There- 
fore putting  S  for  the  space  which  the  pulse  would  run  over 
while  a  falling  body  would  acquire  the  velocity  of  the  pulse,  we 
have  ms  :mti ::  h  :S.      Whence 


g,      mnXh      ms-\-snXh     snX^h  ,  . 

o=- =z =: \-n 

ms  ms  ms 

But  we  have  before  found  H  =  ^^^^-      Substituting    H    for   its 

7ns 

value  in  the  preceding  equation,  we  have  S  =  H-fA.     If  then 

we  put  \\-{-h  for  tlie  velocity  of  the  pulse,  the  space  through 

which  a  body  must  fall  to  acquire  that  velocity  will  be     — — 

In  this  expression  h  is  to  be  regarded  as  the  intensity  of  the 
pulse:  it  being  the  length  of  that  column  of  particles  which  must 
be  superadded  to  the  height  of  a  homogeneous  atmosphere,  in 
order  to  produce  in  the  air  that  degree  of  increased  condensation 
which  the  pulse  effects  in  the  particles  through  which  it  passes. 

If  in  the  expression  last  found  we  make  A  =  0,  the  expression 

TT 

becomes  — .     This  is  the  result  arrived  at  by  Newton,  and  which, 

as  we  have  already  remarked,  was  regarded  by  him  and  is  now 
generally  received  as  the  theoretical  formula  for  the  space  through 
which  a  body  must  fall  to  acquire  the  velocity  of  sound.  But  it 
is  evident  from  our  demonstration  that  the  velocity  due  to  that 
space,  instead  of  being  the  velocity  of  any  assignable  pulse  is 
simply  a  limit  below  which  no  pulse  can  be  propagated  in  an 
elastic  fluid  whose  subtangent  is  H.* 

*  Newton's  demonstration  of  this  problem  has  been  regarded  by  several  dis-  . 
tmguished  maihematieians  as  obscure  and  ineonchi.sive.     It  commences  witli  the 
hypothesis  that  a  particle  put  in  motion  b}'  a  pulse  is  accelerated  and  retarded 


15 

A  pulse  which  produces  the  sensation  of  sound  must  produce 
real  motion  in  the  particles  through  which  it  passes.  In  such  a 
pulse  h  must  have  some  finite  magnitude.  Nor  can  that  mag- 
nitude be  by  any  means  the  smallest  that  is  competent  to  pro- 
duce motion  in  the  air;  for  if  such  were  the  fact,  then  the 
slightest  impulse  given  to  the  air  by  a  vibratory  movement, 
even  waving  the  hand  in  it,  should  produce  the  sensation  of 
sound.  The  intensity  of  a  pulse  which  is  competent  to  produce 
that  sensation,  will  of  course  vary  with  the  sensibility  of  the 
ear  which  is  to  receive  it;  and  consequently  the  nature  of  the 
case  does  not  allow  us  to  assign  any  definite  magnitude  to  the 
minimum  intensity  of  sonorous  pulses;  but  we  know  by  experi- 
ence that  the  velocity  of  the  particles  in  which  a  pulse  originates 
must  be  great  in  order  to  produce  the  sensation  of  sound  in  the 
most  delicate  ear.  We  also  know  that  in  the  case  of  the  heavier 
sounds,  as  the  report  of  cannon,  the  condensation  of  the  particles 
in  which  the  pulse  originates  is  very  intense. 

This  view  of  the  subject  may  throw  some  light  upon  the 
discrepancy  between  the  theoretical  velocity  of  sound,  as  deter- 
mined by  Newton  and  others,  and  its  real  velocity  as  found  by 
experiment.  The  velocity  according  to  that  theory  should  be 
about  944  feet  in  a  second;  varying  slightly  from  this  according 
to  the  state  of  the  barometer,  thermometer,  and  hygrometer. 


The  velocities 
found  by  experi-  \ 
ment,  by 


Roberts, 1300 

Boyle, 1200 

Mersenne, .... 1474 

Flamsteed  and  Halley, 1 142 

Florentine  Academy, 1148 

French  Academy, ..    1172 


according  to  the  law  of  the  oscillating  pendulum.  Gabriel  Cramer  (see  Glasgow- 
edition  of  Xewton's  Principia,  Book  II,  prop.  48,  notes),  objects  to  the  result 
arrived  at  by  Newton,  that  it  flows  from  his  hypothesis  and  not  from  the  nature 
of  things.  To  prove  this  he  deduces  the  same  result  upon  the  hypothesis  that 
the  panicle  is  accelerated  and  retarded  by  a  constant  force.  The  fact  that 
Cramer's  hypothesis  answered  just  as  well  for  the  solution  of  the  problem  as 
Newton's,  seems  not  a  little  to  have  puzzled  the  editor  of  the  edition  of  the 
Principia  referred  to,  who  devotes  several  pages  of  notes  to  the  vindication  of 
Newton's  result  from  its  supposed  bearing.  But  the  enigma  is  solved  when  we 
consider  that  both  Newton  and  Cramer  regard  the  space  through  which  the 
particle  vibrates  as  an  infinitesimal  quantity.  In  such  case,  evidently  it  can  make 
no  difEerence  what  is  assumed  as  the  law  of  acceleration ;  this  being  a  point  at 
whieh  all  laws  coalesce. 


16 

Newton  adopts  the  smallest  of  these  experimental  velocities, 
viz:  1142  feet  per  second,  as  the  true  practical  velocity  of  sound; 
and  to  this  he  reconciles  his  theory;  in  part,  by  a  hypothesis  that 
the  air  consists,  to  a  certain  extent,  of  solid  particles,  through 
which  the  pulse  is  transmitted  instantaneously;  and  in  part  by 
another  hypothesis,  that  the  pulse  does  not  give  motion  to  the 
foreign  matter  which  the  air  contains,  and  so  is  transmitted  so 
much  the  faster  through  the  true  air  as  there  is  less  of  it  in  a 
given  space.  This  explanation  of  the  matter  has  not  been  sat- 
isfactory to  those  who  have  followed  Newton  in  investigating 
this  subject.  They  have  justly  thought  that  such  causes  might 
retard,  but  could  not  accelerate  the  pulse.  Various  other  hy- 
potheses have  been  successively  proposed  and  rejected,  and  a 
vast  amount  of  labor  has  been  exijended  in  the  effort  to  reconcile 
theory  and  practice  in  this  case.  Of  these  hypotheses,  we  Avill 
mention  only  that  which  Laplace  is  said  to  have  regarded  as  the 
true  one,  viz:  the  increased  elasticity  in  the  air  produced  by  the 
heat  evolved  by  condensation.  A  little  reflection  will  serve  to 
show  that  this  cause  also  may  retard,  but  cannot  accelerate  the 
pulse.  In  order  that  the  force  of  the  pulse  may  be  maintained 
without  loss,  the  propelling  force  derived  from  the  reexpan- 
sion  of  the  particles  must  be  not  less  than  their  retarding  force 
in  being  compressed;  and  in  order  to  this,  whatever  heat  is 
evolved  during  their  compression,  must  be  reabsorbed  in  their 
reexpansion.  If,  then,  any  time  is  required  either  for  the  evolu- 
tion of  the  heat,  or  for  its  absorption,  so  that  the  specific  heat 
of  the  particle  does  not  instantaneously  conform  to  the  change 
of  density;  or  if  any  portion  of  the  heat  evolved  is  radiated  and 
lost,  so  as  not  to  be  present  to  be  reabsorbed,  then  the  evolution 
of  heat  must  retard  the  pulse.  Otherwise  it  cannot  affect  the 
velocity  of  its  propagation. 

In  the  view  of  the  subject  we  have  taken,  it  will  cease  to  be 
a  matter  of  surprise  that  the  velocity  of  sound  should  be  found 
to  be  greater  than  that  assigned  by  Newton's  theory;  as,  also, 
that  the  experimental  velocities  should  be  found  to  differ  greatly 
among  themselves,  however  carefully  the  experiments  may  have 
been  tried. 

When  the  velocity  of  a  pulse  is  given,  we  may  find  its  intensity 

V2 

by  the  formula  hz= H.    We  may  find  the  maximum  density, 

Oil 

V« 

the  natural  density  being  1,  by  the  formula,  max.  dens.  =  ^^fj« 


17 

We  may  find  the  space  occupied  by  a  particle  at  the  maximum 
density,  its  natural   extent  being  1,  by  the  formula,  extent   at 

32H 

max.  dens.  =  ^v"„— . 

V  ^ 

When  the  velocity  is  1142  feet  per  second,  if  Ave  put  the  sub- 
tangent  H  =  27818  feet,  we  have, 

Intensity  of  the  pulse,  h  =  12937  feet. 
Maximum  density,  1*465. 
Extent  of  particle,  0*682. 

If  we  take  for  the  given  velocity,  that  Avhich  a  body  Avould 
acquire  by  falling  through  the  subtangent,  or  that  with  Avhich 
air  would  rush  into  a  vacuum,  we  shall  find  h  =  H,  and  the  par- 
ticles will  be  compressed  into  half  their  natural  size,  arid  the 
density  Avill  be  double  the  natural  density. 

A  condensing  force  equivalent  to  the  pressure  of  a  column 
12937  feet  high,  and  which  compresses  the  particles  into  about 
-^-^  of  their  natural  size,  is  a  pulse  "exceedingly  intense"  as 
compared  with  that  which  Newton  supposes;  but  if  our  solution 
of  the  problem  is  correct,  it  is  physically  impossible  that  a  pulse 
of  less  intensity  should  propagate  itself  in  our  atmosphere  with 
a  velocity  of  1142  feet  in  a  second. 


>^    OP   TFTR 

;UNIVBIlSIT7l 

ARTICLE   III. 


On  the  Mode  of  Expansion  of  Elastic  Fluids  as  controlled 
by  Dynamic  Laws* 

That  under  the  controlling  influence  of  the  known  laws  of 
motion,  elastic  fluids  must  expand  according  to  some  definite  and 
invariable  law,  is  an  obvious  truth  and  one  which  has  often  been 
recognized  by  mathematicians.  But  the  determination  of  that 
law  is  a  problem  which  hitherto,  it  is  believed,  has  not  been 
solved.  There  are  many  interesting  points  in  mechanics  and 
physics,  in  relation  to  which  the  present  state  of  knowledge  is 
imperfect,  which  depend  for  their  correct  and  complete  develop- 
ment, in  part  at  least,  on  a  solution  of  this  problem.  It  is  there- 
fore a  point  of  some  interest  to  science.  It  is  our  purpose  in  this 
article  to  solve  this  problem;  and  we  shall  do  so  by  employing 
a  method  similar  in  part  to  that  employed  in  solving  the  problem 
of  the  propagation  of  pulses  in  elastic  media,  in  the  American 
Journal  of  Science,  second  series,  vol.  v,  page  372. 

Before  entering  upon  the  investigation  we  will  here  state  one 
curious  and  remarkable  fact  which  the  investigation  discloses. 
We  revert  to  it  hert  because  a  fact  so  much  at  variance  with  pre- 
conceived notions  may  be  interesting  to  those  readers  who  will 
not  care  to  follow  out  the  mathematical  details  of  this  article. 

When  a  fluid  passes  by  free  expansion  from  one  state  of  density 
to  another,  we  should  naturally  suppose  that  it  must  pass  through 
all  the  intermediate  states  of  density  that  can  be  assigned  between 
the  two.  Such  appears  to  have  been  the  notion  of  every  writer 
who  has  made  reference  to  this  point;  and  at  first  view  it  would 
seem  absurd  to  suppose  that  the  fact  could  be  otherwise.     But 

*  Published  A.  D.  1850,  in  the  Amerioan  Journal  of  Science,  second  series,  vol. 
ix,  p.  334. 


20 

such  is  not  tlie  way  in  which  elastic  fluids  expand.  On  the  con- 
trary the  parts  of  the  fluid  successively  and  inMantcweoiisly 
change  their  density,  to  the  extent  of  one-half  (when  free  to 
expand  to  that  extent)  without  passing  into  the  intermediate 
slates.  As  vapor  is  thrown  off  from  the  surface  of  water  in  a 
tenuous  state  ab  initio,  and  without  having  first  passed  into  those 
states  of  density  which  are  intermediate  between  the  density  of 
the  w  ater  and  that  of  the  vapor,  so  a  column  of  rarefied  fluid  is 
thrown  of  from  the  front  of  a  denser  colunin;  each  infinitesimal 
element  of  the  highest  order  of  the  denser  column,  being  success- 
ively and  instantaneously  transformed  to  the  more  rare  state. 
And  as  the  change  of  density  is  instantaneous,  so  likewise  the 
entire  velocity  due  to  that  change  is  imparted  instantaneously  to 
each  element  successively. 

But  to  proceed  with  the  investigation;  suppose  a  straight  tube 
of  uniform  calibre  extending  indefinitely  in  both  directions  from 
a  given  point.  Suppose  the  tube  on  one  side  of  this  point  to  be 
filled  with  a  column  of  fluid  of  the  density  D,  indefinitely  expan- 
sible, and  always  maintaining  the  same  ratio  between  its  density 
and  elastic  force  when  it  expands;  and  suppose  the  other  portion 
of  the  tube  a  perfect  vacuum.  It  is  required  to  determine  the 
law  according  to  which  the  fluid  expands  into  the  vacuum;  so 
that  we  may  be  able  to  assign  the  precise  state,  of  the  fluid,  in 
respect  to  density  and  velocity,  at  each  and  every  point  of  the 
tube  after  the  lapse  of  any  given  time  from  the  commencement 
of  the  expansion. 

Since  the  elastic  force  is  always  as  the  density,  D  may  repre- 
sent both  the  density  and  the  elastic  force.  The  force  D  acts 
during  the  first  instant  in  every  part  of  the  column,  and  in  every 
direction;  and  therefore  during  that  instant  every  part  of  the 
column  is  kept  in  equilibrio  except  the  first  element.  Conse- 
quently in  the  first  instant  expansion  takes  place  in  the  first 
element  only,  and  as  the  w^hole  force  D  acts  during  that  instant, 
the  parts  of  this  element  must  receive  such  velocities  that  the 
sura  of  their  momenta  shall  be  equal  to  that  due  to  the  action  of 
the  constant  force  D  during  that  time.  It  is  obvious  that  the 
termination  of  the  first  instant  coincides  with  the  commencement 
of  motion  in  the  second  element;  also  that  motion  will  not  com- 
mence in  the  second  element  until  the  density  in  front  of  it  has 
been  to  some  extent  reduced.     Let  the  ratio  in  which  it  is  reduced 

before  motion  begins  in  the  second  element  be  represented  l)y      . 


21 

Then  the  density  of  the  posterior  part  of  the  expanded  element 
at  the  end  of  the  first  instant  is  —     Now  for  reasons  which  will 

X 

soon  be  apparent,  all  the  other  parts  of  the  expanded  element, 
whatever  may  be  their  present  state  of  density,  may  be  con- 
sidered as  having  passed  first  into  the  density  —     But  at    the 

same  time  that  the  grade  —  began  to  form  in  front  of  the 
column,  that  grade  itself  must  have  begun  to  expand  again  in 
the  same  ratio,  forming  another  grade  —  •      And  at  the  same 

time  that  the  grade  — ^  began  to  form,  that  likewise  must  have 

begun  to  expand  in  the  same  ratio  forming  a  grade    — ,  ?    and   so 

on  ad  infinitum.  The  grades  therefore  will  correspond  to  the 
terms  of  an  infinite  series,  in  decreasing  geometrical  progression. 
All  of  them  originate  simtdtaneouslii  in  the  first  element;  and 
yet  every  grade  respectively  may  be  considered  as  having  passed 
into  and  out  of  all  the  grades  which  precede  it ;  inasmuch  as  each 
in  its  origin  is  a  constituent  part  of  that  which  precedes  it.  The 
fluid  which  passes  into  any  one  of  these  grades  in  the  first  instant 
does  not  all  of  it  pass  into  the  next  in  the  same  time;  for  equal 
quantities  by  measure  expand  in  equal  ratios  in  equal  times;  and 
since  a  given  quantity  by  measure  in  any  one  grade  becomes  a 
larger  quantity  by  measure  when  expanded  into  the  next  grade, 
a  portion  will  have  been  left  at  the  end  of  the  first  instant  in  each 
grade  which  has  not  expanded  into  the  next.  Hence  at  the  end 
of  the  first  instant  the  first  element  of  the  column  will  have  been 
distributed  into  portions  or  grades,  having  their  respective  densi- 
ties corresponding  to  the  terms  of  the  infinite  series 

D    D    D    I) 

x^'  ^'  a;"  ic"  ^^^*  ^    ' 

If  we  extend  this  series  backward  one  term,  we  obtain  the 
series 

^    D    D    D    D 

^'    x'  ^'  ^'  x''  ^^''-  ^^^ 

tK'  wj  *Kf  ftt/ 

Since  equal,  quantities  by  measure  pass  out  of  each  of  thesg 
states  in  a  given  time,  if  s  be  the  space  occupied  by  the  original 
4 


22 

element,  and  if  we  multiply  each  of  the  terms  of  the  series  (B) 
by  s,  then  the  terms  of  the  resulting  series 

^     D«    Ds    Ds    D« 

Ds,  — ,  —5-,  —5-,  — J-,  etc.  (C) 

X        X         X         X 

will  severally  express  the  quantities  of  fluid  that  expand  from 
each  grade  respectively  into  the  next.  Now  since  fluids  expand- 
ing in  equal  ratios  acquire  equal  velocities,  equal  velocities  are 
acquired  in  each  of  these  expansions.  If  then  we  find  that 
velocity  and  by  it  multiply  the  sum  of  the  series  (C),  the  product 
will  be  the  sum  of  the  momenta  generated  in,  or  imparted  to,  the 
parts  of  the  first  element  in  the  time  in  which  the  point  of  expan- 
sion recedes  through  s. 

If  the  quantity  Ds  be  expanded  from  the  density  J)  to  the  den- 
sity — ,  the  space  it  will  occupy  will  be  increased  in  the  inverse 

ratio  of  these  densities;  and,  therefore,  — :D :  :s:  sx.    Hence  s  and 

X 

sx  are  respectively  the  spaces  occupied  by  the  element  before  and 
after  the  first  expansion.  Now  the  velocity  which  the  mass  D« 
receives  in  this  expansion,  is  obviously  that  which  would  carry  it 
over  the  difference  between  these  spaces  in  the  time  in  which  the 
expansion  takes  place;  that  is,  the  velocity  imparted  in  the  first 
expansion  is  sx — s:=8.x—l;  and  the  same  velocity  is  imparted  in 
every  other  expansion.  If,  then,  we  multiply  the  sum  of  the 
series  (C)  by  s.x—\,  the  product  will  be  equal  to  the  sum  of  all 
the  momenta  generated  in  the  parts  of  the  element.  This  pro- 
duct is  Ds^x.  Therefore  Ds^x  is  the  entire  amount  of  momentum 
which  the  force  D  is  competent  to  generate  in  the  time  in  which 
the  point  of  expansion  recedes  through  s. 

We  will  now  proceed  to  find  another  expression  for  the  mo- 
mentum which  the  force  D  is  competent  to  generate  in  the  same 
time,  in  order  that  by  comparing  it  with  that  just  found,  we  may 
ascertain  the  value  of  x. 

Let  H  be  the  height  of  a  column  of  fluid  of  the  density  D, 
whose  weight  is  equal  to  the  elastic  force  D;  and  let  II  —  A  be 
the  height  of  another  column  of  the  same  density  whose  weight 

is  equal  to  the  elastic  force  — .     Then  —  :D:  :H:H— A.    Let  mw 

^  XX 

be  the  space  occupied  by  the  first  element  at  the  density  D, 

?u  n  s  . 

I  I  I 


23 

« 

and  ms  that  which  it  occupies  when  expanded  to  the  density—. 
Since  the  spaces  occupied  by  the  element  in  these  states  are 
inversely  as  the  densities,  mn  :  ))ts  ::--:])::  H— A  :  H,  and  there- 

fore,  ijis—.^n  -.uis:  :  II— A:  11:  whence  we  obtain  II  =  .     In 

the  time  in  which  the  point  of  expansion  recedes  through  inn, 
the  element  Ds  receives  a  velocity  which  will  carry  it  over  sn  in 
the  same  time.  If,  then,  7n)i  represent  the  velocity  of  the  point 
of  expansion,  sn  will  represent  the  velocity  imparted  to  the  fluid 
by  the  first  expansion.  Consequently,  the  retrogressive  velocity 
of  the  point  of  expansion  must  be  such  that  in  the  time  in  which 

it  passes  over  any  space,  the  force  D may  give  to  all  the  fluid 

in  that  space  the  velocity  sn.     Hence  the  point  of  expansion  will 

run  over  h  in  the  time  in  which  the  force  D will  give  to  all 

the  fluid  in  h  the  velocity  sn.     But  the  force  D is  equal  to 

the  loeight  of  all  the  fluid  in  A.  Therefore  the  point  of  expansion 
runs  over  A  in  the  time  in  which  the  mass  A  would  in  falling  by 
its  own  gravity  acquire  the  velocity  sn.  The  time  in  which  a 
falling  body  acquires  the  velocity  S7i  is  to  that  in  which  it  would 
acquire  the  velocity  of  the  point  of  expansion,  or  inn  as  sri  to  ?//«/ 
and  the  spaces  over  which  the  point  of  expansion  would  run  in 
these  times  are  in  the  same  ratio.  Therefore,  putting  S  for  the 
space  which  the  point  of  expansion  w^ould  run  over  while  a  falling 
body  would  acquire  the  velocity  of  the  point  of  expansion,  we 
have  s)i  :  mn :  :  A :  S,  or,  sn :  ms  —  sn  :  :  A  :  S;  whence  we  obtain 

hXr"S      ,       r,   ^  1  1    +•        4.-        ^   Ti      f'iXrns      ,„. 

b= A.      Hut  we  have  before  lound  H= .      Ihere- 

sn  sn 

fore  S=II— A;  that  is,  the  point  of  expansion  will  run  over  H— A 

in  the  time  in  which  a  falling  body  will  acquire  the  same  velocity. 

Consequently  the  velocity  of  the  point  of  expansion  is  that  which 

XT I 

a  body  will  acquire  by  falling  through . 

If  the  force  D  act  on  the  mass  H  during  the  time  that  mass 
would  fall  through  H,  it  would  give  that  mass  a  velocity  which 
would  carry  it  over  2H  in  the  same  time,  because  the  force  D  is 
equal  to  the  weight  of  the  mass.  The  mean  velocity  of  a  body 
falling  through  H  is  that  w  hich  will  be  acquired  by  falling  through 


24 

— .      If,  then,  the  point  of  expansion  moved  with  the  velocity 

H  .      ' 

acquired  by  a  body  in  falling  through  — ,  in  the  time  of  passing 

over  H,  the  force  D  would  be  competent  to  give  to  all  the  fluid 
in  H  a  velocity  which  would  carry  it  over  2H  in  the  same  time; 
and,  consequently,  in  the  time  of  passing  over  s  it  would  give  to 
the  mass  l)s  a  velocity  which  would  in  the  same  time  carry  it 
over  25.     But  the  point  of  expansion,  as  before  shown,  moves 

with  the  velocity  acquired  by  falling  through  — - — .     Xow  the 

,     .       ,  H .  ^       ^  H-A         /H         /H-A 

velocity  due  to  — -  is  to  that  due  to  — —  as  4/  -j^^  \/  — x — ;  and 

the  times  in  which  the  point  of  expansion  would  move  over  s 

T    H 
with  these  velocities,  are  inversely  as  these  velocities,  or  as        — 


/i 


1 


to        — ^-.     But  D:—:  :H:H— A,  and  therefore  these  times  are 
2  X 

as        —  to        — ,  or  as  2  to  a/2x.     The  velocitv  which  the  force 
4  2x  ^ 

D  can  impart  in  these  times  is  as  the  times  respectively.     And 

since  it  has  been  shown  that  in  the  former  of  these  times  the 

velocity  25  will  be  imparted  by  the  force  D,  we  have  2',^2x;i 

2s\/2x 
2s', —:=sW2x.     That  is,  the  velocity  which  the  force  D   is 

competent  to  impart  to  the  mass  Ds  in  the  time  in  which  the 
point  of  expansion  recedes  through  s,  is  5y^2.»'.  Consequently 
the  momentum  which  the  force  D  can  imjjart  in  the  same  time  is 
Ds*  a/2.1'.  But  we  have  before  found  this  momentum  to  be  Ds^x. 
Therefore  \)s^x—\)s^  ^2x,  whence  x=y/2x  and  a'=2. 

Having  thus  found  the  absolute  value  of  a*,  if  we  substitute 
this  value  for  x  in  the  series  (A)  we  shall  have,  for  the  densities 
of  the  several  parts  or  grades  into  which  the  first  element  will 
have  been  distributed  at  the  end  of  the  first  instant,  the  respect- 
ive terms  of  the  following  series,  viz: 

D  D  D  D    D 

2'T'  8'  16'  64' 

We  found  the  velocity  of  the  point  of  expansion  to  be  that 


25 

which  a  body  will  acquire  hy  falling  through  — — ;  the  value  of 

H-A     H 

h  being  dependent  on  the  value  of  x.     But  when  a;=:2,  — —  =— . 

Therefore  the  absolute  velocity  of  the  point  of  expansion  is  that 

TT 

which  a  body  will  acquire  by  falling  through  — ,  or  one-fourth  of 

the  subtangent  of  the  fluid. 

Since  the  extent  of  the  element  is  doubled  by  the  first  expan- 
sion, the  velocity  of  the  first  grade  will  be  equal  to  the  velocity 
of  the  point  of  expansion,  or  that  due  to  one-fourth  of  the  sub- 
tangent;  and  an  equal  additional  velocity  is  imparted  in  each 
succeeding  expansion.  If,  then,  v  represent  the  velocity  due  to 
one-fourth  the  subtangent  of  the  fluid,  the  absolute,  velocities  of 
the  several  grades  respectively  will  be  expressed  by  the  respect- 
ive terms  of  the  series  v,  2v,  3y,  4u,  5v,  etc. 

Since  one  element  =s  by  measure  passes  from  each  grade  into 
the  next,  and  becomes  =2s  in  the  next,  the  length  of  each  grade 
at  the  end  of  the  first  instant  =2s—s=iS.  That  is,  the  length  of 
each  grade  is  equal  to  that  of  the  original  element;  and  the  place 
of  the  first  grade  is  that  which  was  occupied  by  the  original 
element,  the  other  grades  succeeding  it  in  continuous  order. 

Having  now  ascertained  the  state  of  things  at  the  end  of  the 
first  instant,  let  us  inquire  what  takes  place  in  the  second  instant. 

It  is  obvious  that  during  the  second  instant  tlie  front  of  the 
second  element  of  the  column,  and  also  the  front  of  each  grade 
respectively  is  a  point  of  expansion  from  which  one  element  =.s 
by  measure  passes  into  the  next  grade.  Thus  in  the  second 
instant  each  grade  receives  an  addition  of  2s  to  its  rear  and  loses 
Is  from  its  front.  The  same  takes  place  in  every  succeeding 
instant.  Since  the  increment  of  the  length  of  the  grades  for  each 
instant  is  s,  the  velocity  of  the  increase  is  v.  The  length  of  the 
grades  is  therefore  always  equal  to  the  space  through  which  the 
point  of  expansion  has  receded  in  the  column.  Thus  while  the 
length  of  the  grades  increases  with  the  uniform  velocity  v,  their 
number,  velocity  and  density  remain  unchanged.  Consequently 
no  other  gradations  of  density  can  exist  in  front  of  a  column 
expanding  into  a  vacuum,  but  those  which  correspond  to  the 

terms  of  the  infinite  geometrical  series  — ,  — ,  — ,  — ,  etc.:  and  no 

2'   4'   8'  16  ' 

other  gradations  of  velocity  but  those  which  correspond  to  the 

terms  of  the  infinite  arithmetical  series  v,  2v,  8y,  4w,  etc. 


26 

The  point  of  expansion  in  the  colunm  recedes  with  the  velocity 
v;  and  since  the  length  of  the  first  grade  is  always  equal  to  the 
space  through  which  that  point  of  expansion  has  moved,  it  fol- 
lows that  the  point  of  expansion  from  the  first  to  the  second 
grade  is  stationary.  And  since  the  second  grade  increases  in 
length  with  the  velocity  v,  the  third  point  of  expansion  moves 
forward  with  the  velocity  v;  and  since  all  the  other  grades 
increase  in  length  with  the  same  velocity  v,  the  velocities  of  the 
several  points  of  expansion  will  be  expressed  by  the  following 
series — v,  0,  v,  2v,  Sv,  4w,  etc. 

In  order  to  give  a  synopsis  of  the  results  to  which  we  have 
come,  let  AB  be  a  column  of  fluid  of  the  density  T),  expanding 
into  a  vacuum  toward  C.  Let  the  velocity  due  to  a  height  equal 
to  one-fourth  of  the  subtangent  of  the  fluid  be  v.  Suppose 
expansion  to  have  commenced  at  B,  and  the  point  of  expansion 
to  have  receded  to  any  distance  ?i.  Set  off  from  B  an  infinite 
number  of  spaces  B«,  ah,  be,  cd,  etc.,  each  equal  to,  B;?.  Then 
the  points  n,  B,  a,  h,  c,  d,  etc.,  are  the  plac^  of  the  points  of 
expansion,  and  the  boundaries  of  the  several  grades,  or  parts 
having  different  degrees  of  density  and  velocity,  into  which  the 
original  mass  Bn  has  been  distributed. 

A  n         B  «  &  c  (Z  etc. 


C^ 


Between  these  points  respectively  the  densities  are 

D    D    D     D     D 

2  ■  4  ■  8  *  16  *  64  ■    ^  ^' 
The  velocities  are 

y  .  2u.  3w.  4y .  5y .  etc. 

These  points  move  toward  C  with  the  velocities 

— V,  0,  V,  2u,  3v,  4y,  etc., 

and  relatively  to  each  other,  and  to  the  fluid,  with  the  velocities 

V,  V,  V,  V,  W,  V. 

As  corollaries  from  the  preceding  investigations  we  may  state 
the  following  propositions: 

1.  No  other  gradations  of  density  can  exist  in  front  of  a  col- 
umn of  fluid  which  is  expanding  toward  a  vacuum  except  tliose 
which  are  found  by  successive  divisions  of  the  original  density 
by  2. 


27 

2.  The  change  of  density  in  the  fluid  in  passing  from  one  of 
these  grades  to  the  next  is  not  gradual  but  instantaneous;  so 
that  .the  grades  are  constantly  separated  from  each  other  by  a 
mere  imaginary  plane. 

3.  No  other  velocities  can  exist  among  the  parts  of  a  fluid 
which  is  expanding  toward  a  vacuum  but  such  as  are  multiples 
of  the  velocities  which  a  body  will  acquire  by  falling  through 
one-fourth  of  the  subtangent  of  the  fluid. 

4.  The  velocity  imparted  to  the  particles  of  an  expanding  fluid 
is  not  the  result  of  a  continual  and  gradual  acceleration,  but  of 
successive  instantaneous  increments  equal  to  that  which  a  body 
will  acquire  by  falling  through  one-fourth  of  the  subtangent  of 
the  fluid. 

It  now  remains  to  consider  the  mode  of  expansion  when  the 
fluid  is  not  free  to  expand  indefinitely,  but  has  its  expansion 
arrested  at  some  given  density  d. 

It  is  obvious  that  if  d  correspond  in  value  to  any  of  the  terms 
of  the  series,  the  manner  of  expansion  up  to  that  point  will  be 
the  same  as  if  the  expansion  were  continued  indefinitely.  There 
will  therefore  be  in  the  expanding  fluid,  in  such  case,  so  many 
grades  corresponding  to  the  terms  of  the  series,  as  there  are  of 
complete  terms  intervening  between  D  and  d.  But  let  us  inquire 
what  takes  place  when  d  does  not  correspond  to  any  terms  of  the 
series.  First,  suppose  d  to  be  greater  than  the  first  term.  Then 
from  what  has  been  before  shown,  the  velocity  of  the  point  of 
expansion  i*  that  which  a  body  will  acquire  by  falling  thi-ough 

TT  L 

when    H  -  h  is  the  height  of  a  column  whose  weight  is 

equal  to  the  elastic  force  of  the  expanded  fluid;   also  that  the 

LT 

velocity  of  the  point  of  expansion  is  that  due  to  the  height  — 

when  the  expansion  is  from  D  to  — .      These    velocities    are    as 

4/  —  to  ^  — - —   and  since  H  :  II  —  A  : :  D  :  f/,    those  velocities 

are  as  4/  —  to  a/  —.     The  velocity  due  to  —  is  v.      Hence  we 

have  \/  -y  '•  \/  —  '  '■  V  :  va/  -^  =z  velocity  of  the  point  of  expan- 
sion in  this  case. 

Let  us  next  find  the  velocity  of  the  fluid.     The  times  of  run- 


28 

niiig  over  s  by  the  point  of  expansion,  with  the  velocities  4/  — 

and  |/  -   are  inversely  as  these  velocities  ;  and  the  velocities 

imparted  to  the  mass  D.s-  in  these  times  are  as  the  products  of 
the  times  by  the  respective  forces.     When  the  velocity  of  the 

point  of  expansion  was  1/-7-  the  force  was  —  and  the  velocity  of 

the  fluid  was  v.     The  force  in  the  present  case  is  D  — f?.     Hence 
D 

~2       D-d  B-d         ,     . 

we  have  — -^  :  — — - : :  y  :  vy'2  .  —jf^  =  velocity  of  the  fluid  in 

VT   Vj  ^ 

this  case. 

Secondly,  suppose  the  value  of  d  to  fall  between  any  two  con- 
secutive terms  of  the  series.  It  is  obvious  that  we  have  now 
only  to  substitute  in  the  expression  last  found  that  term  of  the 
series  which  is  next  greater  than  d  for  D,  and  it  will  then  express 
the  acceleration  due  to  expansion  from  the  last  complete  term 
into  the  fractional  grade. 

To  find  the  retrogressive  velocity  of  the  point  of  expansion, 
relatively  to  the  fluid,  in  the  grade  which  precedes  the  fractional 
grade,  we  must  make  the  like  substitution  of  the  last  complete 

/2(l 
term  for  D  in  the  quantity  u^/  ^  found  above.  The  retrogres- 
sive velocity  of  the  point  of  expansion  in  the  grade  which  pre- 
cedes the  fractional  grade  is  greater  than  in  the  other  grades,  and 
of  course  that  grade  will  be  shorter  than  the  others  in  the  same 
ratio.  This  is  the  only  modification  which  a  fractional  grade 
produces  in  those  that  precede  it.  In  all  other  respects  the  mode 
of  expansion,  up  to  the  fractional  grade,  corresponds  to  the  view 
presented  in  the  foregoing  synopsis. 

We  are  now  prepared  to  construct  a  formula  for  the  final 
velocity  of  a  fluid  which  expands  from  any  density  D  to  any 
other  density  d. 

Let  V  be  the  final  velocity;    v  the  velocity  due  to  a  height 

equal  to  one-fourth  the  subtangent  of  the  fluid;  ji  the  number  of 

.      D    D    D    D  ^.  ^  . 

complete  terms  or  the  series  — ,  — ,  — ,  — ,  etc.,  which  intervene 

between  D  and  d.      Then   u»   is  obviously  the  velocity  of  the 
grade  which  precedes  the  fractional  grade,  if  there  be  a  fractional 


29 

grade.     When  the  first  grade  is  fractional  we  found  its  velocity 

to  be  V a/2. — TT^j'y  and  we  also  found  that  to  suit  this  expres- 

sion  to  the  case  of  a  fractional  grade  occurring  elsewhere  in  the 
range  of  the  series,  we  are  to  substitute  for  D  that  term  of  the 
series  which  is  next  greater  than  d.     Now  the  value  of  that  term 

will  be  — .     Making  the  substitution  accordingly,  the  expression 

for  the  additional  velocity  due  to  expansion  into  the  fractional 

grade  becomes,  after  reducing  wy'2  .  -         yy-      ^J  adding  this 

quantity  to  vn  we  obtain  the  final  velocity  of  the  fluid,  resulting 
from  its  expansion  from  any  density  D  to  any  other  density  d. 
Hence  the  formula  is 


,      /^    D  — 2"(7 

When  there  is  no  complete  term  of  the  series  between  D  and  d, 
71  =  0  and  the  above  formula  becomes 

When  there  is  no  fractional  grade,  that  is,  when  d  is  equal  to 
some  term  of  the  series,  that  part  of  the  formula  beyond  n  equals 
0,  and  then  the  above  formula  becomes  V=:  vn. 

From  the  general  principles  here  developed  it  is  obvious  that, 
as  in  expansion,  so  likewise  in  condensation,  the  transition  of 
an  elastic  fluid  from  one  density  to  another  is  not  by  gradations 
which  may  be  represented  by  a  curve,  but  abrupt,  instantaneous, 
per  saltum  vel  saltus.  Pulses,  therefore,  which  are  propagated  in 
elastic  fluids  partake  of  the  same  character;  that  is,  the  condensa- 
tion and  subsequent  reexpansion  of  the  successive  elements 
through  which  the  wave  moves  is  instantaneous.  This  fact  was 
not  known  when  the  article  on  the  propagation  of  pulses,  referred 
to  at  the  commencement  of  this  article,  was  written.  It  however 
does  not  affect  the  validity  of  the  reasoning  in  that  article. 


ARTICLE  IV. 


Experimental  Demonstration  of  the  Law  of  the  Flow  of 
Elastic  Fluids  which  was  deduced  theoretically  in 
Article  I.*  . 

Ix  volume  V,  second  series,  of  the  American  Journal  of 
Science,  page  78,  I  proposed  a  new  theory  of  the  flow  of  elastic 
fluids  through  orifices,  differing  essentially  from  that  heretofore 
received.  The  chief  object  of  the  present  article  is  to  give  an 
account  of  an  experiment  instituted  for  the  purpose  of  testing 
the  truth  of  that  theory. 

The  fundamental  points  of  difference  between  the  old  theory 
and  the  new,  are  as  follows: 

1.  The  old  theory  regards  the  constant  force  which  expels  the 
fluid  as  being,  in  all  cases,  equal  to  the  difference  between  the 
elastic  forces  of  the  fluids  in  the  two  vessels. 

The  new  theory  regards  it  as  equal  to  that  difference  only 
when  the  less  exceeds  half  the  greater;  and  in  all  other  cases  as 
equal  to  half  the  greater.  . 

2.  The  old  theory  considers  the  fluid  as  passing  the  orifice  with 
a  density  equal  to  that  in  the  discharging  vessel. 

The  new  theory  considers  it  as  passing  the  orifice  with  a 
density  equal  to  that  in  the  receiving  vessel,  whenever  this  last 
is  equal  to  or  greater  than  half  the  density  in  the  discharging 
vessel;  and  in  all  other  cases  with  half  the  density  in  the  dis- 
charging vessel. 

The  formula  for  the  quantity  discharged  in  a  given  time, 
predicated  upon  the  new  theory,  gives,  in  all  cases,  less  than 

*  Published  A.  D.  1851,  iu  the  American  -Journal  of  Science,  second  series, 
Vol.  xii,  p.  186. 


32 


that  predicated  upon  the  old  theory.     In  tlic  case  of  a  flow  into 
a  vacuum,  the  difference  amounts  to  one-half. 

The  scheme  devised  to  test  the  relative  merits  of  the  two 
theories,  was  founded  upon  the  following  considerations,  viz: 
When  air  rushes  from  the  atmosphere  into  a  receiver  wholly  or 
partially  exhausted,  passing  on  its  way  through  a  small  inter- 
mediate vessel  or  chamber,  entering  that  chamber  and  passing 
out  of  it  through  equal  orifices,  it  will  take  in  that  chamber  a 
density  somewhere  intermediate  between  that  of  the  atmosphere 
and  that  in  the  receiver.  For  each  relation  that  may  at  any 
moment  subsist  between  the  density  of  the  atmosphere  and  that 
in  the  receiver,  thf  density  in  the  chamber  will  have  a  certain 
definite  and  determinate  value,  such  that  the  chamber  may 
receive  through  one  orifice  and  discharge  through  the  other 
simultaneously  the  same  quantity  of  air.  Nom'  since  in  order  to 
this  equal  simultaneous  flow  the  two  theories  respectively  demand 
quite  different  densities  in  the  chamber,  the  object  of  ray  experi- 
ment was  to  ascertain  the  actual  densities  in  such  a  chamber 
under  various  relations  of  the  density  in  the  receiver  to  the 
density  of  the  atmosphere,  in  order  to  compare  the  densities  thus 
ascertained  experimentally  with  those  demanded  by  each  theory 
respectively  in  like  circumstances. 


To  try  the  experiment,  I  constructed  the  apparatus  shown  in 
figure  1.  A  i-^  a  vessel  or  receiver  of  tlie  capacity  of  about 
fifty    gallons,    so    arranged    that    it    may    be    exhausted    l)y   the 


83 

air-pump  or  otlierwise.  B  is  an  elbow  formed  of  lead  pipe  of 
one  inch  calibre,  one  branch  of  which  opens  into  the  receiver, 
and  the  end  of  the  other  branch  at  C  is  covered  by  a  brass  plate 
or  disc  about  ^ig^th  of  an  inch  in  thickness,  through  which  is  an 
orifice  of  about  jig-th  of  an  inch  in  diameter.  Another  similar 
plate  with  an  orifice  of  the  same  size  intersects  the  pipe  at  D, 
thus  forming  a  chamber  between  the  two  plates.  Two  short 
tubes  are  inserted  into  the  lower  side  of  the  pipe;  one  on  each 
side  of  the  plate  D.  With  these  short  tubes  two  glass  tubes  in 
and  »,  each  thirty-tliree  inches  in  length,  are  connected  by  means 
of  pieces  of  India  rubber  hose.  These  glass  tubes  are  open  at 
both  ends  and  terminate  at  the  bottom  in  a  vase  of  mercury,  A 
rod  {not  shown  in  the  sketch)  graduated  to  inches  and  tenths  is 
placed  beside  the  glass  tubes,  sustained  upon  a  float  resting  upon 
the  surface  of  the  mercury,  so  adjusted  that  zero  of  the  gradua- 
tion may  coincide  with  the  surface  of  the  mercury. 

If  the  orifice  at  C  be  closed  by  a  stopper  and  the  receiver 
exhausted,  the  mercury  will  rise  in  the  tubes;  and  if  the  density 
of  the  atmosphere  at  the  time  of  the  experiment  be  expressed  in 
inches  of  mercury,  the  height  of  the  mercury  in  the  tubes  as 
read  upon  the  graduated  rod  will  be  equal  to  the  difference 
between  the  density  of  the  atmosphere  and  that  in  the  receiver. 
If  we  now  remove  the  stopper  from  the  orifice  at  C,  the  column 
of  mercury  in  the  tube  m  will  instantly  subside  to  a  point  which 
indicates  the  difference  between  the  density  of  the  atmosphere 
and  the  density  in  the  chamber  when  an  equal  quantity  of  air 
flows  through  the  two  orifices;  while  at  the  same  time  the  column 
of  mercury  in  the  tube  ?i  will  only  have  begun  to  subside  very 
slowly  as  the  density  in  the  receiver  increases.  Having  noted 
the  height  of  the  barometer  at  the  time  of  the  experiment,  if  we 
note  the  simultaneous  heights  of  these  two  columns  of  mercury, 
and  deduct  them  respectively  from  the  height  of  the  barometer, 
we  shall  have  the  density  in  the  chamber  necessary  to  an  equal 
flow  through  the  two  orifices  under  the  relation  which  subsists 
at  the  moment  of  notation  between  the  density  in  the  receiver 
and  the  density  of  the  atmosphere.  And  if  we  note  the  simul- 
taneous heights  of  these  columns  at  various  times  during  the 
filling  of  the  receiver,  so  many  densities  in  the  chamber  shall  we 
find  corresponding  to  the  different  relations  of  the  other  two 
densities. 

At  the  time  of  the  exi^eriment  the  height  of  the  barometer,  or 


34 

density  of  the  atmosphere  was  thirty  inches.  In  consequence  of 
leaks  in  the  receiver,  I  was  unable  to  exhaust  it  so  as  to  raise  the 
column  in  the  tube  w  higher  than  twenty-six  inches.  I  noted 
the  simultaneous  altitudes  of  the  two  columns  at  the  moment 
when  the  column  n  coincided  with  each  successive  inch-mark 
upon  the  graduated  rod,  and  thence  ascertained  the  densities  in 
the  chamber  under  twenty-six  different  relations  between  the 
density  in  the  receiver  and  that  of  the  atmosphere.  These 
results  I  have  placed  in  the  table  beyond,  in  w^hich  the  first 
column  shows  the  densities  in  the  receiver  at  the  times  of  nota- 
tion, and  the  second  the  densities  in  the  chamber  corresponding 
thereto. 

In  order  to  ascertain  what  these  densities  should  have  been 
according  to  the  old  theory,  I  constructed  a  formula  as  follows: 
Let  /I  be  the  height  of  the  barometer  at  the  time  of  the  experi- 
ment, D  the  density  in  the  receiver,  d  the  density  in  the  cham- 
ber, V  the  velocity  through  the  first  orifice,  v  the  velocity 
through  the  second  orifice.  Then  according  to  the  old  theory 
the  force  which  drives  the  air  through  the  first  orifice  is  A  —  d 
and  that  which  drives  it  through  the  second  orifice  is  c?— D. 
But  since  an  equal  quantity  flows  through  both,  these  forces  are 
as  the  velocities,  that  is  A  —  d  :  d—T>  :  :  V :  y. 

Again,  according  to  the  old  theory  the  density  with  which  the 

air  passes  the  first  orifice  is  A,  and  that  with  which  it  passes  the 

second  orifice  is  d.     But    since  the  orifices  are  equal  and  the 

quantities  which  j^ass  through  them  are  also  equal,  the  products 

of  the  velocities  by  the  densities  are  equal,  that  is  AY^dv  and 

dv 
V=— .     Substituting  this  value  of  V  in  the  preceding  couplet 

and  then  finding  the  value  of  d,  we  have  the  following  formula 
for  determining  the  densities  in  the  chamber  according  to  the 
old  theory,  viz  : 


d=.\/ 


4  2      * 


The  several  densities  in  the  chamber  computed  by  this  formula 
are  placed  in  the  fourth  column  of  the  table. 

In  order  to  ascertain  what  the  densities  in  the  chamber  should 
have  been  according  to  the  new  theory,  I  constructed  a  formula 
as  follows,  preserving  the  same  notation  as  above. 

By  the  new  theory  the  force  which  drives  the  air  through  the 


35 

first  orifice  is  A  —  d  whenever  d  is  not  less  than  — .     But  d  is 

2 

never  less  than  —  when   an  equal   quantity  flows  tlirough  both 

orifices,  for  if  it  were  so  the  chamber  would,  according  to  our 
theory,  be  receiving  as  much  as  could  flow  into  a  vacuum  under 
the  pressure  A,  and  must  therefore  discharge  into  the  receiver 
as  much  as  would  flow  into  a  vacuum  under  a  pressure  A;  in 
order  to  which  the  density  in  the  chamber  must  be  equal  to  A, 

and  therefore  greater  than  --.     Consequently,  the  force  which 

drives  the  air  through  the  Jirst  orifice  is  in  this  arrangement 
always  A  —  d.      Again,  the  force  expended  in  driving  the  air 

through  the  second  orifice  by  the  new  theory  is  —  whenever  D 

d 
is  not  greater  than  — .     Let  us  first  construct  a  formula  for  the 
2 

cases  in  which  D  is  not  greater  than  — .  In  these  cases  the  den- 
sities  under  which   the  air   passes  the  orifices  are  respectively 

d 

A—d  and  — •.     Since  the  forces  are  as  the  velocities, 

2  ' 

and  since  the  quantities  are  equal,  dY^=—-,  and  V=— .      Substi- 

4 
tuting  this  value  of  V  in  the  couplet,  we  have  d=^—^;  a  constant 

5 

quantity.     Hence  while  the  density  in  the  receiver  varies  from 

2 
0  to  —  A,  the  density  in  the  chamber  is  a  constant  quantity  and 

4 
equal  to  —  A.     Let  us  now  construct  a  formula  for  finding  the 

d 
value  of  d  when  D  is  greater  than  — .     In  these  cases  the  forces 

are  A—d  and  d—  D  and  we  have  for  the  couplet  A—d:d—D:  :Y:v. 

The  densities  in  the  orifice  are  d  and  D  and  we  have  dY=T)v 

Dv 
and  Y=---.    Substituting  this  last  quantity  in  the  couplet  we  find 


■V  D' 


J-Bl'     A-D 

+  -T^+-2- 


36 


as  the  formula  for  the  value  of  d  hy  the  uew  theory  when  D 

2 
exceeds  —  J.     The  densities  in  the  chamber  computed  by  these 

formulae  are  placed  in  the  third  column  of  the  table. 

Density  of  the  Athosphebe  during  Experiment  30. 


Density 

In  the 

Receiver. 

Henslty  in  the 

chamber  as  found 

by  experiment. 

Density  in  the 
chainbeV  due  to 
the  new  theory. 

Density  in  the 
chauibiT  due  to 
the  old  theory. 

Deviation  of  the 
experiment  from 
the  new  theory. 

0 

24 

18.541 

~ 

I 



24 

18.820 



2 



24 

19.106 



3 



24 

19.397 



4 

24.58 

24 

19.695 

.58 

5 

24.58 

24 

20. 

.58 

6 

24.58 

24 

20.31 1      ■ 

.58 

7 

24.53 

24 

20.628 

.58 

8 

24.58 

24 

20.953 

.58 

9 

24.58 

24 

21.284 

.58 

]0 

24.58 

24 

21.622 

.58 

11 

24.60 

24 

21.968 

.60 

12 

24.64 

24 

22.320 

.64 

13 

24.70 

24.032 

22.713 

.668 

14 

24.77 

24.124 

23.048 

.646 

15 

24.89 

24.270 

23.423 

.62 

16 

25.03 

24.464 

23.805 

.566 

17 

25.21 

24.700 

24.194 

.51 

18 

25.43 

24.974 

24.594 

.456 

19 

25.69 

25.280 

25. 

.41 

20 

25.96 

26.615 

25.414 

.345 

21 

26.30 

25.976 

25.835 

.324 

22 

26.65 

26.360 

26.265 

.29 

23 

27.03 

26.764 

26.699 

.266 

24 

27.43 

27.186 

27.149 

.244 

25 

27.81 

27.625 

27.604 

.185 

26 

28.20 

28.076 

28.066 

.124 

27 

28.65 

28.541 

28.533 

.109 

28 

29.08 

29.017 

29.016 

.063 

29 

29.55 

29.504 

29.504 

.046 

30 

30. 

30 

30. 



The  affinity  of  the  experimental  results  to  those  derived  from 
the  new  theory,  is  obvious  upon  inspection  of  the  table;  and  the 
want  of  affinity  to  those  derived  from  the  old  theory,  is  not  less 
evident.  The  comparative  relation  of  the  two  theories  to  the 
results  of  experiment,  is  more  readily  seen  in  the  annexed  cut  (fig. 
2),  where  they  are  respectively  delineated  by  a  curve.  The  upper 
curve  represents  the  densities  or  elastic  forces  in  the  chamber, 
as  found  by  experiment;  the  next  curve  those  due  to  the  new 
theory,  and  the  lower  curve  those  due  to  the  old  theory. 

Notwithstanding  the  near  approximation  of  the  exi)erimental 


be 


mm 
mm 


imi 


m 

■H 


■■■L 

■■■■»■■■ 


■■■■■HHB 


IH 


\ 


38 


results  to  those  due  to  t\e  new  theory,  there  is  yet  a  small  but 
distinct  deviation,  which 'nholds  throughout.  This  deviation  in- 
dicates either  that  there  i^  some  cause  affecting  the  flow  which 
the  theory  does  not  take  ir^io  account,  or  that  in  the  structure  of 
the  apparatus  or  in  trying%^e  experiment,  there  was  some  failure 
to  comply  with  the  requisil^^  conditions. 

Although  the  apparatus/  as  rude  in  its  structure,  yet  care  had 
been  taken  to  secure  a  cogwliance  with  the  conditions  on  which 
the  experiment  was  base^f  and  in  conducting  the  experiment  I 
was  assisted  by  my  friend.  Prof.  A.  C.  Twining,  a  gentleman 
distinguished  for  his  accuracy  in  such  matters.  The  experiment, 
moreover,  was  several  times  repeated,  with  no  important  differ- 
ence in  the  results.  For  these  reasons,  in  seeking  the  cause  of 
the  deviation,  my  first  inquiry  was  whether  it  might  be  attri- 
buted to  a  change  in  the  ratio  of  elastic  force  to  density;  the 
theory  being  predicated  upon  the  assumption  that  this  ratio  is 
constant.  It  has  been  ascertained  by  experiment,  that  when  air 
is  condensed  and  then  suffered  to  lose  the  heat  evolved  by  con- 
densation, the  ratio  of  its  elastic  force  to  its  density  will  be 
diminished.  Hence  it  is  certain  that  a  part  or  the  whole,  or 
possibly  even  more  than  the  whole  heat  evolved  by  condensation 
will  be  required  to  prevent  that  ratio  from  being  diminished. 
Still,  however,  it  has  generally  been  assumed  by  philosophers 
(I  know  not  on  what  grounds)  that  if  air  is  suddenly  condensed, 
so  as  not  to  allow  the  heat  evolved  by  condensation  to  escape, 
the  ratio  of  elastic  force  to  density  will  be  increased.  This 
assumption  was  made  by  •  Laplace  when  he  attributed  to  this 
cause,  in  part,  the  velocity  of  sound.  Let  us  suppose  then,  for 
the  present,  that  in  sudden  condensation  the  ratio  of  elastic  force 
to  density  is  increased.  It  will  then  follow  that  in  sudden  ex- 
pansion, the  ratio  of  elastic  force  to  density  will  be  diminished. 
But  if  that  ratio  were  diminished,  then  the  deviation  in  the  table 
should  be  in  the  opposite  direction;  that  is,  the  experimental 
results,  instead  of  being  greater  than  the  theoretical,  should  be 
less.  The  deviation,  therefore,  is  not  accounted  for  by  this  sup- 
position; on  the  contrary,  the  experiment  seems  to  prove  that 
the  ratio  is  not  diminished  by  expansion,  and  therefore  cannot 
be  increased  by  condensation,  as  Laplace  supposed. 

Let  us  next  take  the  contrary  supposition,  Viz:  that  the  ratio 
of  elastic  force  to  density  is  increased  by  expansion.  This  would 
cause  a  deviation  in  the  same  direction  as  we  find  in  the  table. 


39 

In  order  to  ascertain  whether  the  deviation  in  question  is  due  to 
this  cause,  we  must  next  inquire  whether  a  deviation  arising 
from  this  cause,  would  vary  in  the  same  manner  throughout  the 
table,  as  does  the  observed  deviation.  Now  if  we  go  through 
the  table  and  assign  for  each  observation  severally,  the  manner 
in  which  the  ratio  of  elastic  force  to  density  must  increase,  in 
order  to  satisfy  that  observation,  we  shall  find  very  nearly  one 
and  the  same  increment  of  the  ratio  demanded  for  all  the  obser- 
vations. Hence  if  we  attribute  the  deviation  to  this  cause  we 
should  be  obliged  to  conclude  that  one  and  the  same  change  in 
the  ratio  takes  place,  whether  the  expansion  be  greater  or  less. 
But  such  a  conclusion  is  obviously  inadmissible.  We  cannot, 
therefore,  attribute  the  deviation  in  question  to  a  change  in  that 
ratio,  either  by  increase  or  diminution. 

Nor  can  we  ascribe  the  deviation  to  that  which  is  the  chief 
cause  of  deviation  from  theory  in  the  case  of  the  flow  of  liquids, 
viz:  the  contraction  of  the  stream  in  passing  an  orifice.  For  if 
that  cause  operated,  it  would  affect  the  flow  in  the  same  ratio  in 
both  orifices,  and  therefore  would  not,  in  this  case,  affect  the 
indications  of  the  mercurial  columns.  Moreover,  I  think  it  can 
be  shown,  from  consideration  a  2)^'iori,  that  the  cause  which 
produces  the  contraction  of  the  stream  in  liquids,  could  not 
operate  to  affect  the  flow  of  expansible  fluids. 

Having  satisfied  myself  that  the  deviation  was  not  due  to  the 
(Causes  above  named,  my  next  inquiry  was,  whether  a  difference 
in  the  sizes  of  the  orifices  (hitherto  assumed  to  be  equal)  would 
cause  a  deviation  corresponding  to  that  in  the  table.  In  examin- 
ing this  point,  I  found  that  the  experimental  results  would  be 
very  nearly  satisfied  throughout  the  table,  by  the  assumption 
that  the  area  of  the  second  orifice  was  less  than  that  of  the  first, 
in  about  the  ratio  of  .933  to  1.  As  the  two  orifices  had  been 
made  as  nearly  equal  as  they  could  be  by  forcing  the  same  steel 
plug  through  both,  I  was  confident  that,  as  originally  formed, 
they  could  not  differ  to  this  extent.  But  it  occurred  to  me  that 
some  accidental  circumstance  might  have  occurred  to  diminish 
the  inner  orifice,  and  I  suspected  that  the  workman,  in  handling 
the  brass  plate  after  the  orifice  was  made,  had  got  dirt  into  it, 
and  had  omitted  to  cleanse  it  before  soldering  on  the  outer  plate. 
To  ascertain  whether  such  was  the  case,  I  divided  the  tube  near 
the  second  orifice,  and,  upon  examining  it  with  a  microscope, 
discovered  that  there  was  dirt  adhering  around  its  inner  periph- 


40 

ery  sufficient,  I  think,  to  cause  a  diminution  of  its  area  to  the 
extent  above  named.  Unfortunately  this  discovery  was  made 
after  the  arrangements  for  trying  the  experiment  had  been 
removed;  and  I  have  not  since  found  leisure  to  replace  them  and 
try  the  experiment  anew.  But  for  this  accidental  circumstance 
no  doubt  there  would  have  been  a  still  nearer  approximation  of 
the  experimental  results  to  those  derived  from  the  formula.  The 
coincidence,  however,  is  sufficiently  near  to  Establish  the  truth 
of  the  new  theory,  so  far  as  respects  those  points  of  difference 
between  the  two  theories  specified  in  the  first  part  of  this  article. 

The  fifth  column  of  the  table  shows  the  several  differences 
between  the  experimental  results,  and  those  due  to  the  new 
theory.  It  will  be  noticed  that  these  differences  increase  slightly 
between  density  10  and  13  in  the  receiver,  before  they  began  to 
decrease.  This,  I  think,  indicates  a  slight  obstruction  to  the 
flow  through  the  second  orifice,  when  the  density  in  the  receiver 
becomes  equal  or  nearly  equal  to  that  of  the  effluent  stream. 
This  increment  at  its  maximum  amounts  to  .088,  corresponding 
to  the  pressure  of  that  portion  of  an  inch  of  mercury,  and  is,  I 
think,  the  measure  of  the  obstruction  or  resistance  due  to  that 
circumstance.  If  this  view  of  the  subject  is  correct,  then  there 
would  have  been  a  deviation  to  this  extent  in  this  part  of  the 
table,  even  if  the  orifices  had  been  equal. 

It  is  desirable  that  further  experiments  of  this  kind  should  be 
tried  by  those  who  have  better  means  at  command  than  I  had  to 
do  justice  to  the  subject.  To  such  as  may  be  disposed  to  under- 
take it,  I  would  suggest  that  a  perfect  equality  of  the  two  orifices 
might  be  secured  by  interchanging  the  discs,  varying  the  sizes  of 
the  orifices  until  they  gave  the  same  indications  in  both  positions. 

If,  after  thus  securing  the  equality  of  the  orifices,  there  should 
still  be  a  deviation  in  that  part  of  the  table  where  the  elastic 
force  in  the  chamber  is  constant,  such  deviation,  I  think,  must  be 
attributed  to  a  change  in  the  ratio  of  elastic  force  to  density; 
and  if  so,  its  amount  would  furnish  the  means  of  determining 
the  law  according  to  which  that  ratio  varies. 

I  would  also  suggest  that  a  modification  of  this  experiment 
would  furnish  perhaps  the  best  possible  means  of  determining 
the  law  according  to  which  the  ratio  of  elastic  force  to  tempera- 
ture varies,  when  the  absolute  amount  of  heat  is  constant.  In 
an  arrangement  for  this  purpose,  the  bulb  of  a  thermometer 
should  be  inserted  into  the  chamber,  and  the  outer  orifice  should 


41 

be  so  constructed  that  it  may  be  enlarged  or  diminished  at  pleas- 
ure. With  this  arrangement,  we  may  cause  the  air  in  the  cham- 
ber to  assume,  almost  instantly,  any  elastic  force  we  may  choose, 
between  the  elastic  force  of  the  atmosphere,  and  a  little  more 
than  twice  the  elastic  force  in  the  receiver;  and  we  may  keep 
that  force  constant  in  the  chamber  during  any  time  that  may  be 
required  to  cool  the  thermometer  down  to  the  corresponding 
temperature,  the  continual  flow  through  the  chamber  in  the  mean- 
time carrying  off  not  only  the  heat  which  flows  in  from  extrane- 
ous sources,  but  also  that  derived  from  the  thermometer  itself. 
We  may  thus  ascertain  the  relation  of  elastic  force  to  temperature 
at  as  many  points  as  we  please  within  this  range,  and  thereby 
determine  the  law  of  their  variation  when  the  absolute  amount 
of  heat  remains  constant. 


ARTICLE  V. 


The  Form,  Formation  and  Movement  of  Sonorous  Waves* 

There  can  be  no  doubt  that  the  movements  of  aerial  particles 
resulting  from  forces  impressed  upon  them,  and  the  transmission 
of  force  from  one  to  another  are  in  perfect  accordance  with 
Dynamic  laws.  But  the  various  attempts  which  have  been  made 
to  determine  by  the  application  of  these  laws  the  precise  order 
and  extent  of  their  respective  movements  have  not  been  success- 
ful. While  by  the  application  of  Dynamic  laws  to  other  matter 
we  are  able  to  trace  with  the  utmost  precision  the  paths  of  bodies 
in  the  remotest  regions  of  space,  we  have  as  yet  no  definite 
knowledge  respecting  the  movements  of  the  air  which  surrounds 
us; — not  even  of  those  movements  on  which  we  are  constantly 
dependent  for  the  transmission  to  each  other  of  our  oral  com- 
munications. There  are  many  problems  pertaining  to  this 
department  of  Physics  the  solution  of  which  would  be  of  great 
interest  to  science,  but  to  solve  which  no  attempts  have  as  yet 
been  made  because  of  the  apparent  difficulties  involved  in  the 
investigation.  Whatever  of  thought  has  been  bestowed  in  this 
field  of  inquiry  has  been  devoted  chiefly  or  wholly  to  the  deter- 
mination of  the  velocity  of  sound.  But  the  attempts  to  solve 
even  this  problem,  initiated  by  Newton,  have  been  attended  with 
so  little  success  that  now,  after  the  lapse  of  two  centuries,  during 
which  continual  efforts  have  been  made  to  reconcile  the  widely 
differing  results  of  theory  and  experiment,  scientists  are  not 
agreed  whether  this,  or  that,  or  both,  require  the  correction 
which  should  bring  them  into  harmony. 

*  Read  at  a  meeting  of  the  Connecticut  Academy  of  Arts  and  Sciences,  Decem- 
ber 21,  1881. 


44 

The  delay  to  develop  this  branch  of  Physics  has  been  due,  no 
doubt,  to  the  peculiar  and  complex  nature  of  aerial  matter;  its 
tenuity,  its  fluidity,  its  compressibility  and  its  elasticity.  These 
properties  have  seemed  to  scientists  to  conspire  to  embarrass  the 
application  of  Dynamic  laws  to  the  air;  while  its  total  invisi- 
bility has  compelled  them  to  pursue  their  investigations  in  the 
dark,  by  abstract  thought  alone,  and  without  the  aid  of  the 
senses  to  check  the   aberrations  of  reason. 

On  account  of  these  properties  of  aerial  matter  the  problem 
to  find  the  velocity  of  sound  has  always  been  regarded  by  mathe- 
maticians as  one  of  great  difficulty.  The  late  Professor  Peirce, 
in  his  Treatise  on  Sound,  says,  "  The  problem  to  investigate  the 
general  laws  of  the  propagation  of  sound  is  one  of  the  utmost 
complexity,  and  has  been  resolved  only  under .  very  restricted 
conditions."  After  alluding  to  the  profound  researches  which 
have  been  bestowed  on  this  subject  by  Euler,  Lagrange  and 
others,  he  proceeds  to  investigate  the  problem  restricted  to  the 
case  of  propagation  in  a  tube  of  uniform  calibre.  After  conr 
structing  an  elaborate  series  of  differential  equations  to  this 
end,  he  at  Ifength  arrives  at  one  which  he  says  "  is  altogether 
intractable  and  incapable  of  integration."  In  order  to  proceed 
further  he  finds  it  necessary  to  restrict  the  conditions  of  the 
problem  still  further  by  confining  it  to  the  case  in  which  the 
elastic  force  of  the  wave  exceeds  that  of  the  medium  in  which  it 
is  propagated  by  an  infinitesimal  quantity  only;  that  is,  by  con- 
fining it  to  a  sound  of  infinitesimal  intensity.  With  the  problem 
thus  restricted  he  goes  on,  and  after  a  very  elaborate  process, 
arrives  at  the  conclusion  that  the  velocity  of  sound  is  that  which 
a  heavy  body  will  acquire  by  falling  through  half  the  height  of 
a  homogeneous  atmosphere.  This  is  the  same  conclusion  as  that 
arrived  at  by  Newton,  Euler,  D.  Bernouilli,  Lagrange,  Poisson, 
Laplace,  and  I  know  not  by  how  many  others,  all  of  whom  found 
it  necessary  to  restrict  the  problem  to  like  conditions. 

When  nature  is  questioned  respecting  her  laws,  even  by  the 
most  profound  mathematicians  and  by  the  most  recondite  pro- 
cesses of  analysis,  if  those  processes  are  based  upon  assumptions 
not  consonant  with  her  laws  she  will  refuse  to  respond  except 
in  equations  "  intractable  and  incapable  of  integration."  It  is 
the  purpose  of  this  communication  to  show  that  those  who  have 
attempted  to  solve  this  problem  have  made  an  erroneous  assump- 
tion respecting  the  manner  in  which  an  impinging  force  imparts 


46 

motion  to  aeinal  matter,  and  that  upon  a  correction  of  this  error 
the  difficulties  which  they  encountered  in  their  attempts  to  give 
a  general  solution  to  the  problem  will  vanish,  and  we  shall  be 
able  to  arrive  at  such  a  solution  in  a  direct  and  simple  way. 

The  mistake  to  which  I  have  referred  consists  in  having  as- 
sumed that  motion  is  imparted  by  a  finite  force  to  the  infinit- 
esimal particles  of  aerial  matter  in  the  same  manner  as  to  finite 
masses  of  matter;  that  is  by  acceleration:  or,  in  other  words, 
that  in  the  former  as  well  as  in  the  latter  case  motion  begins 
with  an  infinitesimal  velocity  which  is  augmented  to  a  finite 
velocity  by  infinitesimal  increments.  Such  a  notion  is  not  con- 
sistent with  Dynamic  laws.  These  laws  demand  that  when  a 
force  of  finite  magnitude  acts  upon  an  infinitesimal  quantity  of 
matter  a  finite  velocity  proportionate  to  the  force  should  be 
imparted  in  an  infinitesimal  time.  An  impinging  force  therefore 
does  not  impart  motion  to  aerial  matter  by  acceleration. 

But  (it  may  be  asked  here),  has  not  Newton  in  the  4'7th  pro- 
position of  the  Second  Book  of  the  Principia,  demonstrated  that 
each  infinitesimal  particle  of  air  that  is  put  in  motion  by  a  sono- 
rous wave  is  accelerated  and  retarded  according  to  the  law  of  the 
vibrating  pendulum? 

It  is  true  that  the  proposition  referred  to  purports  to  demon- 
strate this;  but  a  careful  examination  will  show  that  it  fails  to 
do  it.  In  this  proposition  Newton  at  the  outset  assumes  in  his 
hypothesis  the  point  to.  be  proved  and  the  conclusion  at  which 
he  arrives  results  from  that  assumption.  It  is  in  short  a  notable 
example  of  "reasoning  in  a  circle."  This  will  be  evident  upon 
a  close  scrutiny  of  the  argument:  and  it  is  also  clearly  shown  by 
Gabriel  Cramer*  who,  to  prove  the  inconclusive  character  of 
Newton's  reasohing  in  this  proposition,  shows  that  by  a  precisely 
similar  course  of  reasoning  {et,  mutatis  mutandis,  in  totidem 
verbis)  it  may  be  made  to  appear  that  the  law  of  acceleration  is 
that  which  pertains  to  a  uniform  force,  as  in  the  case  of  bodies 
falling  by  their  own  gravity. 

We  may  conclude  then  that  it  has  not  been  demonstrated  by 
Newton  that  the  particles  of  air  put  in  motion  by  a  sonorous 
wave  acquire  their  velocity  by  acceleration;  and  therefore  we 
may  accept  the  conclusion  to  which  we  were  led  by  Dynamic 
laws,  that  each  particle  as  successively  encountered  receives  its 
full  velocity  instantaneously. 

*  See  Principia,  Glasgow  Edition,  page  273,  note. 

1 


i6 


Let  us  now  consider  the  modus  operandi  by  which  the  force  so 
imparted  is  passed  along  from  particle  to  particle;  and  to  facili- 
tate our  conceptions  of  the  process  let  us  suppose  it  to  take 
place  in  a  tube  of  uniform  calibre.  Let  PL,  Fig.  1  be  such  a 
tube,  and  let  P  be  a  piston  fitted  to  it. 


Fig.  ]. 


If   a   uniform  velocity  v  be  instantaneously  imparted   to  the 
piston,  the  plate  of  air  which  is  in  contact   with  its  front   Avill 
instantaneously  receive  the  velocity  of  the  piston  and  at  the 
same  time  be  condensed  by   a  force  equal  to  that  required  to 
overcome  the  inertia  of  the  plate.     Suppose  the  velocity  r  to  be 
such  that  this  condensing  force  is  sufficient  to  reduce  the  thick- 
ness of  the  plate  ^;  then  will  this  plate,  thus  reduced  in  thick- 
ness, be  added  on  to  the  front  of  the  piston   and  go  on  with   it, 
condensing  the  next  plate,  giving  it  the  velocity  v  and  pushing 
it  on   in  front  of  itself,  and  so  on.     Thus  it    will  be  seen   that 
when  the  condensing  force  is  that  here  supposed,  the  point  where 
the  condensation  takes  place  moves  forward  in  the  tube  just  ten 
times  as  fast   as  the   piston.     When  therefore  the   piston   has 
moved  -^  of  an  inch  there  will  be  j^  of  an  inch  of  condensed  air 
in  front  of  it  moving  with  the  same  velocity  as  the  piston.     If 
now,  at  this  juncture,  the  motion  of  the  piston  be  arrested,  the 
condensed   air  in  front  of   it   will   continue  to  move  on   by   its 
momentum  condensing  and  adding  plates  to  its  front  as  before, 
and  in  the  meantime  the  rear  end  of  this  condensed  air  having 
advanced   beyond   the   piston   will   have  space  to  expand    and  • 
resume  its  normal  volume  and  come  to  rest:  and  as  it  respects 
each  infinitesimal  plate,  the  restoration  will  be  as  instantaneous 
as   was   its  condensation.      That   such    will   be  the  manner  of 
restoration  need  not  be  shown  here,  since  it  will  fully  appear 
from  what   is  shown  in  an  article  on  the  mode  of  Expansion  of 
P^lastic  Fluids  in  the  American  Journal  of  Science,  second  series, 
vol.  ix,  page  034.*    Thus  we  have  a  self-sustaining,  self-propagat- 
ing wave,  the  quantity  condensed  in  front  in  a  given  time  being 
just  equal  to  the  quantity  expanded  in  the  rear  in  the  same  time; 
the  latter  by  its  reaction  furnishing  the  power  to  keep  up  the 
motion  and  condense  the  plates  in  front. 

*  The  third  of  the  preceding  articles. 


4t 

A  wave  then,  in  the  sense  in  which  we  shall  employ  the  term 
in  this  paper,  consists  of  a  quantity  of  air  of  a  uniform  density 
greater  than  that  of  the  medium  in  which  the  wave  is  propagated, 
and  having  the  absolute  velocity  that  is  due  to  the  action  of  the 
force  which  effected  its  increased  condensation. 

It  may  be  well  here  to  define  a  few  other  terms  or  phrases, 
some  of  which  we  may  have  occasion  to  employ  in  a  sense  some- 
what different  from  that  attached  to  them  by  other  writers  on 
this  subject. 

The  breadth  of  a  icave  is  the  space  occujiied  by  the  condensed 
air  measured  in  the  direction  in  which  it  moves. 

The  intensity  of  a  wave  is  the  excess  of  its  elastic  force  or 
density  over  that  of  the  medium. 

The  velocity  of  a  icave  is  the  velocity  of  the  point  where  the 
particles  are  condensed  and  put  in  motion. 

The  velocity  of  the  particles  is  the  absolute  velocity  instanta- 
neously impressed  upon  the  particles  successively  by  the  wave. 

In  the  use  of  the  terra  particles  we  are  not  to  be  understood  as 
indicating  any  theory  respecting  the  constitution  of  the  atmos- 
phere,— we  mean  small  parts  or  portions, — not  molecules  or 
atoms. 

In  showing  how  a  wave  may  be  formed  in  a  tube  we  have  sup- 
posed the  motion  of  the  piston  to  be  arrested  after  moving  ^  of 
an  inch.  If  it  had  been  arrested  after  moving  any  smaller  dis- 
tance the  breadth  of  the  wave  would  have  been  smaller  in  the 
same  ratio,  but  in  every  other  respect  it  would  have  been  the 
same;  it  would  have  had  the  same  intensity,  the  same  velocity, 
and  it  would  have  imparted  the  same  velocity  to  the  particles; 
these  quantities  being  dependent  only  upon  the  velocity  of  the 
piston,  and  not  on  the  distance  through  which  it  moved  before 
its  motion  was  arrested. 

We  have  thus  shown  how  a  self -propagating  wave  may  be 
formed  in  a  tube.  The  process  of  the  formation  of  such  a  wave 
in  the  open  air  is,  in  principle,  precisely  the  same,  and  the  form 
of  the  wave  and  laws  of  its  propagation  are  the  same. 

It  is  well-known  that  sounds  of  small  intensity  propagated 
through  tubes  have  the  range  of  their  audibility  extended  to  a 
greater  distance  than  it  could  reach  if  propagated  in  the  open 
air;  but  this  function  of  the  tube  does  not  involve  any  modifica- 
tion of  the  laws  of  propagation  on  which  the  velocity  of  the 
wave  depends.     The  reason  why  a  tube  has  this  effect  may  be 


48 

readily  illustrated.  Suppose  we  present  the  end  of  a  tube  to  a 
vibrating  body  or  other  agent  that  may  originate  a  wave;  then 
a  portion  of  the  condensed  air  which  constitutes  the  wave  at  its 
origin  will  enter  the  tube,  constituting  a  self-propagating  Avave 
therein;  the  rest  of  it  will  spread  out  forming  a  spherical  shell 
or  wave  whose  thickness  measured  on  the  radius  of  the  sphere  is 
the  breadth  of  the  wave;  and  since  there  can  be  no  increase  in 
the  quantity  of  this  condensed  air  after  the  wave  is  formed,  the 
thickness  of  the  shell  or  breadth  of  the  wave  will  be  inversely  as 
the  square  of  the  distance  from  the  place  of  its  origin.  Thus  we 
see  that  the  breadth  of  the  wave  which  is  i)ropagated  outside  of 
the  tube  diminishes  in  a  rapid  ratio;  but  the  breadth  of  that 
which  is  propagated  inside  of  the  tube  undergoes  no  diminution 
except  such  as  may  be  due  to  friction  or  imperfect  elasticity. 
Now  it  is  evident  that,  other  things  being  equal,  the  audibility 
of  a  sound  must  be  as  the  breadth  of  the  wave;  and  that  a 
sufficient  breadth  of  wave  may  be  maintained  in  the  tube  after 
the  breadth  of  that  outside  shall  have  been  so  far  reduced  as  to 
be  incapable  of  producing  the  required  action  upon  the  acoustic 
organs.  This  is  the  only  difference  there  is  between  waves  pro- 
pagated in  the  open  air  and  those  which  are.  propagated  in  a 
tube,  and  as  we  have  shown,  it  is  not  a  difference  that  affects  the 
laws  of  their  propagation. 

From  the  law  of  the  transmission  of  force  from  particle  to 
particle,  as  we  have  shown  it  to  be,  we  will  proceed  to  deduce 
the  law  of  the  propagation  of  sonorous  waves. 

Prop.  I.  The  velocity  of  a  wave  of  whatever  intensity  is  to 
the  absolute  velocity  which  it  impresses  instantaneously  on  every 
particle  over  which  it  passes,  as  the  space  occupied  by  the  con- 
densed air  before  condensation  to  its  loss  of  space  by  condensa- 
tion. 

The  truth  of  this  proposition  may  be  drawn  as  a  corollary 
from  what  was  shown  in  describing  the  formation  of  a  wave  in 
a  tube,  and  therefore  we  need  to  give  no  further  proof  of  it  here. 

Fig.  2. 


Let  he  fig.  2  be  a  column  of  air  of  the  density  and  elastic  force 
due  to  the  pressure  of  the  atmosphere,  and  whose  length  is  equal 
to  the  height  of  a  homogeneous  atmosphere.     Let  this  column 


49 

be  extended  to  any  distance  a;  and  let  a  wave  whose  elastic 
force  exceeds  the  pressure  of  the  atmosphere  in  the  same  ratio 
as  ac  exceeds  he  pass  over  the  column  ae. 

Prop.  n.  The  velocity  of  a  wave  like  that  above  described 
will  be  to  the  velocity  which  it  impresses  on  the  particles  over 
which  it  passes  as  ac  to  ah. 

For  it  is  evident  that  such  a  wave  will  condense  the  particles 
in  the  ratio  of  ac  to  he,  and  therefore  the  space  occupied  by  the 
condensed  particles  before  condensation  will  be  to  their  loss  of 
space  by  condensation  as  ac  to  ah,  and  therefore  by  Prop.  I  the 
velocity  of  the  wave  is  to  that  which  it  impresses  on  the  j^articles 
as  ac  to  ah. 

Prop.  III.  The  velocity  impressed  upon  the  particles  is  that 
which  a  falling  body  would  acquire  in  the  time  in  which  the 
wave  passes  over  ah. 

For  during  that  time  the  excess  of  elastic  force  whicli  is  a 
constant  force  and  which  is  equal  to  the  weight  of  the  air  in  ah 
is  acting  upon  that  air,  and  therefore  must  impart  to  it  the  same 
velocity  as  it  would  acquire  by  falling  during  the  same  time; 
and  that  velocity  is  the  same  as  would  have  been  acquired  in  the 
same  time  by  any  other  falling  body. 

Prop.  IV.  The  velocity  of  the  wave  is  that  which  a  falling 
body  will  acquire  in  the  time  in  which  the  wave  runs  over  ac. 

For  it  was  shown  in  Prop.  Ill  that  in  the  time  of  running  over 
ah  a  falling  body  would  acquire  the  velocity  which  the  wave 
impresses  upon  the  particles,  and  since  by  Prop.  II  that  velocity 
is  to  the  velocity  of  the  wave  as  ah  to  ac,  it  follows  that  in  the 
time  of  running  over  ac  the  falling  body  will  acquire  the  velocit\' 
of  the  wave. 

Prop.  V.  The  velocity  of  a  wave  is  that  which  a  body  will 
acquire  by  falling  through  a  height  which  exceeds  half  the 
height  of  a  homogeneous  atmosphere  in  the  same  ratio  as  the 
elastic  force  of  the  wave  exceeds  the  elastic  force  of  the  medium 
in  which  the  wave  is  propagated. 

It  was  showTi  in  the  last  proposition  that  in  the  time  in  which 
the  wave  runs  over  ac  a  falling  body  would  acquire  the  velocity 
of  the  wave,  and  since  the  mean  velocity  of  falling  bodies  is  half 
their  final  velocity,  it  follows  that  in  the  time  in  which  the  wave 
runs  over  ac,  the  falling  body  in  acquiring  the  velocity  of  the 
wave,  will  have  fallen  through  a  space  equal  to  half  a  c  ;  and 
since  5  c  is  by  construction  equal  to  the  height  of  a  homogeneous 


atmosphere,  and  since  a  c  exceeds  h  e  in  the  ratio  in  which  the 
elastic  force  of  the  wave  exceeds  that  of  the  air,  it  follows  that 
half  of  a  c  exceeds  half  the  height  of  a  homogeneous  atmosphere 
in  the  same  ratio.  Therefore  the  velocity  of  a  sonorous  wave 
is  that  which  a  body  will  acquire  in  falling  through  a  height 
which  exceeds  half  the  height  of  a  homogeneous  atmosphere  in 
the  same  ratio  as  that  in  which  the  elastic  force  of  the  waAe 
exceeds  the  elastic  force  of  the  medium  in  which  the  wave  is 
propagated. 

Thus  we  have  a  general  solution  of  the  problem  relating  to  the 
propagation  of  sound — a  solution  limited  by  no  conditions. 
From  this  solution  it  appears  that  an  excess  of  elastic  f  wee  above 
that  of  the  medium  is  essential  to  the  existence  of  a  loave  ;  and 
also  that  the  greater  this  excess,  the  greater  will  he  the  velocity  of 
the  wave. 

If  we  apply  this  general  solution  to  the  particular  case  of  a 
wave  whose  intensity  (or  excess  of  elastic  force),  is  so  small  that 
its  effect  on  the  velocity  is  inappreciable,  Me  shall  find  the 
velocity  of  such  a  wave  to  be  that  which  is  due  to  half  the 
height  of  a  homogeneous  atmosphere.  This  is  the  velocity  found 
by  Newton  and  those  who  succeeded  him  in  this  investigation, 
all  of  whom,  in  consequence  of  their  assumption  of  the  gradual 
acceleration  of  the  particles,  found  it  necessary  to  confine  their 
investigations  to  waves  of  this  small  intensity.  It  seems  obvious 
that  a  result  thus  obtained  can  be  legitimately  applied  only  to 
such  waves  ;  but  we  find  that  these  scientists  have  regarded  and 
treated  it  as  equally  applicable  to  all  waves.  They  maintain 
that  the  velocity  which  a  body  acquires  by  falling  through  half 
the  height  of  a  homogeneous  atmosphere  is  the  theoretical 
velocity  of  all  Avaves  so  far  as  their  velocity  depends  on  the  laws 
of  Dynamics. 

I  think  it  may  be  shown  hoM'  those  scientists  have  been  led  to 
this  conclusion  as  a  consequence  of  their  acceptance  of  the  theory 
of  gradual  acceleration.  It  is  evident  that  in  a  wave  originated 
and  propagated  by  gradual  acceleration  (if  such  a  wave  can  exist, 
which  I  do  not  admit),  the  foremost  particles  can  have  only 
an  infinitesimal  condensation,  however  dense  the  particles  may  be 
furthest  back.  Now  it  appears  both  from  the  general  solution 
we  have  hei'e  given  and  from  the  more  limited  one  given  by 
others,  that  particles  of  so  small  condensation  can  only  transmit 
their  force  from  one  to  another  with  the  velocitv  due  to  half  the 


51 

height  of  a  homogeneous  atmosphere  ;  and  since  a  wave  can 
move  no  faster  than  its  foremost  particles  it  would  thence  fol- 
low that  no  wave  would  move  faster  than  with  the  velocity  due 
to  half  the  height  of  a  homogeneous  atmosphere,  and  conse- 
quently that  all  waves  have  that  velocity. 

We  see  then  that  if  we  adopt  the  theory  of  gradual  accelera- 
tion we  cannot  avoid  the  conclusion  that  all  sounds  have  the 
same  velocity.  But  we  shall  see  how  absurd  are  the  results  to 
which  we  should  be  driven  by  accepting  the  conclusion  that  all 
Avaves  have  the  same  velocity. 

According  to  Professor  Peirce  the  theoretical  velocity  of 
sound  is  916  feet  per  second,  and  its  actual  or  observed  velocity 
is  1090  feet  per  second  (Peirce  on  Sound,  page  31).  The  great 
difference  between  these  velocities  was  a  mystery  to  mathema- 
ticians from  the  time  of  Newton  to  that  of  Laplace.  Laplace 
suggested  that  the  difference  was  due  to  the  heat  evolved  by 
condensation.  This  suggestion,  coming  from  so  high  a  source, 
has  been  accepted  by  most  mathematicians  without  examina- 
tion as  an  adequate  supplement  to  the  theory  of  Newton,  and 
sufficient  to  reconcile  the  results  of  theory  with  those  of  obser- 
vation. Others  who  have  investigated  the  question  have 
doubted  its  sufficiency  for  that  purpose.  Professor  Peirce,  after 
remarking  that  the  heat  evolved  by  sudden  condensation  may  be 
much  greater  than  we  should  expect  in  the  case  of  such  small 
condensations  as  are  contemplated  in  the  theory  of  sound, 
accepts  with  manifest  misgivings  and  reluctance  the  sufficiency 
of  Laplace's  explanation,  and  employs  it  to  bridge  over  this 
otherwise  impassable  gulf  in  the  theory  of  the  propagation  of 
sound. 

Now  let  us  see  how  much  the  temperature  of  a  wave  must  be 
raised  by  evolved  heat  to  increase  its  velocity  from  916  to  1090 
feet  per  second.  According  to  Professor  Peirce  the  velocity  of 
sound  increases  0*96  feet  per  second  for  every  degree  of  temper- 
ature above  32°  Fahrenheit.     Then  by  the  formula 

1090  —  916 

we  find  that  181.25  degrees  is  the  elevation  of  temperature 
required  to  increase  the  velocity  of  a  wave  from  916  to  1090  feet 
per  second.  And  since  under  the  theory  of  gradual  acceleration 
the  front  of  every  wave  must  consist  of  particles  only  infinitesi- 


62 

mally  condensed,  and  since  this  part  must  have  its  velocity 
increased  as  much  as  any  other  part,  it  would  follow  that  an  in- 
finitesimal condensation  would  cause  an  evolution  of  heat  that 
would  raise  the  temperature  181  degrees  !  It  would  further  fol- 
low that  we  could  not  converse  with  a  friend,  even  in  the  mildest 
whispers,  without  pouring  into  his  ears  waves  of  a  temperature 
of  181  -1-32  =  213  degrees,  or  hotter  than  boiling  water  !  !  More- 
over it  would  also  follow  that  upon  the  slightest  sudden  change 
of  barometric  pressure  we  should  be  immersed  in  air  of  a  temper- 
ature of  213  degrees  !  ! 

Such  are  some  of  the  absurd  conclusions  which  follow  from  the 
doctrine  of  gradual  acceleration  supplemented  by  evolved  heat. 

The  doctrine  that  the  particles  which  constitute  the  wave 
have  all  their  motion  impressed  upon  them  instantaneously,  leads 
to  no  such  absurd  conclusions  and  presents  no  such  hiatus  to  be 
bridged  over  by  evolved  heat  or  otherwise;  as  we  \\411  now 
attempt  to  show. 

It  follows  from  our  solution  of  the  problem  of  the  propagation 
of  sound  that  the  intensity  of  the  wave  (an  item  which  was  elim- 
nated  in  the  old  theory),  is  a  quantity  not  to  be  disregarded  in 
determining  the  velocity  of  sonorous  waves.  Let  us  see  if  this 
item,  when  given  its  proper  place  in  the  investigation,  is  not 
sufficient  to  effect  a  reconciliation  between  the  results  of  theory 
and  those  of  observation  ;  and  that  without  calling  in  the  aid  of 
evolved  heat. 

Prof.  Peirce  gives  916  feet  per  second  as  the  theoretic  axiA  1090 
as  the  observed  velocity  of  sound.  But  we  should  here  note  the 
significant  fact  that  916  is  the  computed  velocity  of  a  wave  so 
weak  that  its  intensity  may  be  regarded  as  cipher;  while  1090 
is  the  observed  velocity  of  the  report  of  a  cannon,  one  of  the 
most  intense  sounds  known  to  us.  According*  to  both  the  old 
and  the  new  theory,  916  is  the  theoretic  velocity  of  the  weaker 
of  these  waves.  If  then  we  can  show  that  under  the  new  theory 
1090  is  also  the  theoretic  velocity  of  the  report  of  a  cannon,  we 
shall  have  effected  a  full  reconciliation  between  theory  and  ob- 
servation without  calling  in  the  aid  of  evolved  heat.  In  order 
to  do  this  the  question  to  be  solved  is  this, — is  the  intensity  of 
the  wave  produced  by  the  discharge  of  a  cannon  sufficient  to 
account  for  the  difference  between  916,  the  velocity  of  a  wave 
whose  intensity  is  regarded  as  cipher,  and  1090,  the  observed 
velocity  of  the  report  of  a  cannon.     If  we  knew   what   is   the 


53 

intensity  of  the  "wave  so  produced  we  could  readily  ascertain 
whether  it  is  sufficient  for  this  purpose;  but  as  we  have  no  means 
of  learning  this,  let  us  inquire  what  the  intensity  of  such  a  wave 
must  be  in  order  that  its  velocity  may  be  1090  feet  per  second. 
Referring  to  fig.  2  where  he  represents  the  height  and  density  of 
a  homogeneous  atmosphere  and  ah  the  intensity  of  the  wave,  the 
question  to  be  solved  is,  what  must  be  the  ratio  of  ah  to  he  when 
the  velocity  due  to  half  the  height  he  is  916  feet,  and  that  due 
to  half  the  height  ac  is  1090  feet?  By  an  obvious  process,  which 
need  not  be  given  in  detail  here,  it  will  be  found  that  ah  must  in 
such  case  be  about  ^^  of  hc\  that  is,  the  density  of  the  wave 
must  exceed  the  density  of  the  air  by  about  41  per  cent. 

That  the  density  of  the  air  in  front  of  the  muzzle  of  a  cannon 
is  upon  its  discharge  increased  as  much  as  41  per  cent.,  and 
often  very  much  more  than  41  per  cent.,  there  can  be  no  doubt. 
It  is  evident  that  the  degree  of  that  condensation  is  not  always 
the  same,  but  must  depend  on  various  considerations,  such  as 
whether  the  cannon  be  charged  with  shot, — whether  the  wad  be 
so  compressed  as  to  require  great  force  to  drive  it  out, — the 
length  of  the  piece, — tlie  quantity  and  quality  of  the  powder, 
&c.  In  the  records  of  the  various  velocities  observed  no  refer- 
ence is  made  to  these  considerations.  Prof.  Peirce  gives  1090  as 
the  mean  of  seven  selected  observations,  none  of  which  varied 
materially  from  1090.  But  there  are  many  other  observations 
recorded  giving  observed  velocities  much  higher  than  those  which 
he  selected.  These  are  rejected  by  Prof.  Peirce  as  unreliable;  for 
what  reason  is  not  apparent.  For  my  own  part  I  doubt  not  that 
waves  are  often  produced  by  the  discharge  of  cannon  whose 
intensity  is  such  that  under  our  theory  their  velocity  of  propaga- 
tion would  far  exceed  1090  feet  per  second.  If  this  be  so,  we 
shall  need  no  aid  from  evolved  heat  until,  in  the  case  of  any 
particular  wave  it  can  be  shown  that  its  velocity  is  greater  than 
under  our  theory  can  be  due  to  its  intensity.  It  may  be  asked 
here  whether  I  reject  the  notion  that  the  velocity  of  sound  is 
increased  by  evolved  heat?  I  answer  that  I  do  not;  but  I  think 
that  under  the  old  theory  the  effect  of  heat  has  been  greatly 
overrated.  I  think  its  effect  must  be  in  proportion  to  the  quan- 
tity of  heat  evolved,  and  that  the  quantity  evolved  must  be  in 
proportion  to  the  degree  of  condensation.  I  think  therefore  that 
when  the  condensation  is  infinitesimal  the  effect  of  the  evolved 
heat  will  be  inappreciable.      If   this  be  so,  then,  whenever   a 


54 

method  shall  be  devised  whereby  we  may  find  by  experiment  the 
velocity  of  waves  of  infinitesimal  condensation,  w^e  shall  find  it 
not  to  differ  materially  from  that  which  a  body  will  acquire  by 
falling  through  half  the  height  of  a  homogeneous  atmosphere; 
— the  velocity  found  by  Newton. 

Those  who  have  investigated  the  propagation  of  sound  under 
the  old  theory  seem  to  have  supposed  that  the  evolved  heat  has 
a  thermo-dynamical  effect;  that  it  is  converted  into  mechanical 
power  and  so  acts  mechanically  to  increase  the  velocity  of  sound; 
but  it  can  have  no  such  action,  for  the  supposed  force  of  the  heat 
is  expended  within  and  upon  the  very  matter  which  is  supposed 
to  be  propelled  by  it,  and  therefore  acts  backward  as  much  as 
forward.  A  vessel  sails  no  faster  because  she  is  on  fire  in  her 
hold;  the  earth  revolves  no  faster  upon  its  axis  because  of  a  fire 
on  its  surface  or  within  it;  the  heat  of  the  fire  under  the  boiler, 
or  of  the  steam  within  it  does  nothing  to  propel  a  steamer,  until, 
through  the  intervention  of  machinery  it  is  made  to  react  upon 
matter  without.  Archimedes  could  move  the  world,  but  he  must 
have  something  outside  of  the  world  for  his  lever  to  react  upon. 
In  the  case  of  the  heated  wave  there  is  nothing  outside  of  the 
wave  for  the  heat  to  react  upon,  and  therefore  it  can  have  no 
mechanical  effect  to  increase  its  velocity.  The  mechanical  power 
of  a  wave  to  perform  the  work  of  reproducing  itself  is  given  to 
it  by  the  power  which  originates  the  Avave,  and  it  can  neither 
be  increased  or  diminished  by  the  heat  evolved  by  condensation. 
What  then  is  the  modus  operandi  in  which  the  velocity  of  a 
wave  is  increased  by  evolved  heat?  It  is  simply  by  increasing 
the  ratio  of  the  elastic  force  of  the  wave  to  the  density  of  the 
medium  in  which  it  is  propagated.  The  elastic  force  of  the  wave 
is  increased  while  the  density  of  the  medium  is  unchanged.  The 
result  of  this  change  of  ratio  is  that  the  velocity  imparted  to  the 
particles  by  the  wave  is  diminished,  while  the  number  of  parti- 
cles acted  upon  in  a  given  time  is  increased  in  the  same  ratio,  so 
that  the  amount  of  mechanical  work  performed  is  neither  in- 
creased or  diminished  by  the  evolved  heat. 

In  the  book  published  a  few  years  since,  entitled,  ''  The  (.'or- 
relation  of  P^orces"  it  was  stated  that  the  eminent  German  scientist 
Mayer  had  deduced  the  mechanical  equivalent  of  heat  from  the 
velocity  of  sound;  and  with  a  result  coinciding  very  nearly  with 
the  equivalent  found  in  other  ways.  This  statement  was  appar- 
ently regarded  by  the  author  as  being  highly  confirmatory  of 


m 

the  doctrine  of  his  book.  This  computation  by  Mayer,  however, 
could  only  be  founded  on  the  assumption  that  heat  acts  mechani- 
cally in  increasing  the  velocity  of  sound;  and  therefore  it  is 
evident  from  what  has  been  shown  that  whatever  other  ground 
the  doctrine  of  the  correlation  of  forces  may  have  to  rest  upon,  it 
can  derive  no  legitimate  support  from  this  computation  by  Mayer, 
A^^e  have  another  example  of  a  sfmilar  mistake  in  the  recent 
aimouncement  that  another  scientist  has  deduced  the  velocity  of 
sound  from  the  mechanical  equivalent  of  heat. 

I  had  intended  to  present  here  several  other  interesting  con- 
clusions at  which  I  had  arrived  on  this  subject,  but  I  forbear, 
lest  I  should  extend  this  paper  to  too  great  length;  and  I  pass 
them  by  with  the  less  reluctance,  because  any  one  who  will  be 
likely  to  take  the  trouble  to  read  this  paper  will  be  able  himself 
to  arrive  at  the  same  results. 

In  conclusion  I  think  proper  to  mention  the  fact  that  more 
than  thirty  years  since  I  contributed  to  the  Journal  of  Science 
an  article*  in  which  I  demonstrated  in  a  manner  quite  different 
from  that  pursued  in  this  article,  but  not  less  conclusive,  that  all 
waves  have  not  the  same  velocity,  but  that  their  velocities  vary 
with  their  intensity;  and  that  the  velocity  of  a  wave  was  that 
which  a  body  would  acquire  by  falling  through  a  height  greater 
than  half  the  height  of  a  homogeneous  atmosphere  in  the  same 
ratio  as  the  density  of  the  wave  exceeds  the  density  of  the 
medium  in  which  it  is  propagated: — results  precisely  similar  to 
those  deduced  in  this  article. 

That  article  seems  to  have  attracted  little  or  no  attention :  for 
up  to  the  present  moment  all  scientific  periodicals  continue  to 
speak  of  the  velocity  of  sound  as  if  it  were  a  quantity  determin- 
ate and  invariable,  sought  for,  but  not  yet  ascertained  with 
precision  and  certainty;  and  scientists,  by  ingenious  theoretical 
and  experimental  devices  still  continue  to  search  for  the  mythical 
number  as  the  Alchemists  did  for  the  Philosopher's  stone.  Pos- 
sibly this  article  may  have  as  little  effect  as  the  former  to  bring 
those  labors  to  an  early  close;  but  however  this  may  be  I  cannot 
doubt  that  sooner  or  later  the  fact  will  be  recognized  and 
accepted,  that  sound  waves  do  not  all  move  with  one  and  the 
same  velocity;  and  that  the  difference  in  their  velocities  is  due 
chiefly  to  the  difference  in  the  intensities  of  the  forces  by  which 

*  Journal  of  Science,  second  series,  vol.  v,  page  312.  The  second  of  the  pre- 
ceding articles. 


56 

they  are  respectively  originated,  and  only  in  a  much  smaller 
degree  to  the  elevation  of  the  temperature  of  the  wave  by  com- 
pression :  and  it  will  be  seen  that  we  have  no  need  to  resort  to 
the  theory  of  Laplace  to  account  for  the  difference  between  the 
velocity  of  the  wave  formed  by  the  report  of  a  cannon  and  that 
of  a  wave  of  infinitesimal  intensity. 


Answers  to  the  principal  objections  which  have  been  made 
to  the  last  of  the  foregoing  Articles. 


Objection  I. 


In  the  foi'egoing  articles  it  is  maintained  that  sound  waves 
which  diffei'  in  intensity  are  not  propagated  with  the  same 
velocity.  To  this  doctrine  it  has  been  objected  that  when  a 
tune  is  played  at  a  great  distance  from  the  hearer,  the  notes 
reach  his  ear  in  regular  order  and  in  their  proper  sequence  in 
respect  to  time ;  which,  it  is  alleged,  would  not  be  the  case  if 
the  waves  moved  with  different  velocities.  Let  us  inquire  into 
the  force  of  this  objection. 

If  we  consider  the  minute  spaces  through  which  strings,  reeds, 
etc.,  vibrate  in  giving  origin  to  musical  sounds,  and  the  limited 
number  of  vibrations  made  per  second,  whatever  may  be  the 
pitch,  we  shall  not  be  able  to  make  out  that  the  velocity  of  the 
vibrating  string  or  reed  is  in  any  case  greater  than  one  or  two 
feet  per  second.  In  the  computation  upon  which  we  are  enter- 
ing, the  greater  we  assume  that  velocity  to  be  the  more  will  the 
result  favor  the  objection ;  and  as  we  can  afford  to  be  very  liberal 
to  the  objector  we  will  assume  that  the  greatest  velocity  that 
can  pertain  to  any  vibrating  body  that  sends  forth  musical  notes, 
may  be  10  feet  per  second.  The  vibrating  body  imparts  to  the 
air  its  own  velocity  ;  and,  from  what  is  shown  in  Article  V.,  it 
follows  that  in  putting  air  in  motion  with  a  velocity  of  10  feet 
per  second,  its  density  will  be  increased  only  so  far  as  to  exceed 
the  normal  density  by  about  one  per  cent.  Now  suppose  that  in 
the  weakest  waves  that  can  transmit  an  audible  sound  there  is  a 
condensation  of  one-tenth  as  much  :  then  the  densities  of  the 
strongest  and  weakest  sonorous  waves  that  can  be  produced  by 
musical  instruments  will  be  to  each  other  as  101  to  lOO'l,  and 
their  velocities  of  propagation  as  .y/101  to  y'lOOl.  Consequently, 
if  two  waves,  thus  differing  in  density,  be  originated  simulta- 
neously at  a  distance  of  1000  feet  from  the  observer,  while  the 


58 

denser  wave  runs  over  the  1000  feet,  the  less  dense  wave  will 

run  over  "^ — =  995*534  feet,  and  will  then  be  behind 

^101 

the  other  1000— 995'534  =  4-466   feet  in  (h'stanee  ;   and  in  time 

(if   the  velocity  of  the  denser  wave  is  1000  feet  per  second), 

-—-=^^  of  a  second. 

995-534-J-4-466     ^"^ 

Thus  it  appears  that  two  waves  having  as  great  a  difference  of 
intensity  as  can  exist  between  any  two  waves  produced  by  musi- 
cal instruments,  if  originated  simultaneously  at  the  same  place 
and  thence  propagated  1000  feet,  Avill  then  be  separated  by  4-466 
feet  in  space  and  by  ^^  of  a  second  in  time. 

But  the  impression  of  a  musical  note  on  the  ear  is  not  pro- 
duced by  a  single  icave,  but  by  a  series  or  group  of  waves ;  occu- 
pying in  their  production  seldom  or  never  less  than  ^  of  a 
second,  and  therefore  extending  through  a  space  of  i|^=50 
feet.  Consequently  the  two  groups  of  waves,  of  which  the  single 
waves  we  have  been  considering  respectively  constitute  the  first, 
overlap  each  other  and  are  coincident  for  more  than  ^  of  their 
length. 

That  displacements  as  great  as  that  above  considered  do  not 
perceptibly  impair  the  regularity  and  harmony  of  a  tune  is 
evident  from  the  fact  that  in  every  musical  performance  however 
perfect,  there  are  constantly  recurring  deviations  from  the  true 
time  of  the  tune  to  the  extent  of  not  less  than  ^^  of  a  second  ; 
and  such  deviations  must  produce  displacements  of  like  extent, 
even  though  we  assume  that  the  velocities  of  all  waves  are  pre- 
cisely equal. 

Again — in  the  case  of  a  number  of  musicians  performing  in 
concert,  although  we  should  assume  that  all  waves  move  with 
the  same  velocity  and  that  there  is  not  the  slightest  deviation 
from  the  true  time,  yet  if  the  difference  in  the  distances  of  the 
hearer  from  the  respective  performers  should  amount  to  4*466 
feet — say  4^  feet, — there  would  result  an  equal  displacement  of 
the  notes.  1  think  it  will  not  be  claimed  that  the  most  fastidious 
critic  of  musical  performances  could  perceive  a  distortion  of  the 
music  in  such  a  case. 

Our  conclusion  then,  is  that  the  apparent  regularity  in  the 
order  and  chronic  sequence  of  the  notes  of  a  tune,  performed  at 
any  distance  from  which  it  can  be  heard,  does  not  go  to  show 
that  all  sound  waves  are  propagated  with  the  same  velocity. 


59 

Objbctiqx  II. 

Those  who  have  heretofore  attempted  to  investigate  the  law  of  the 
propagation  of  sound  have  assumed  that  a  sonorous  wave  consists 
of  contiguous  particles  of  compressed  air  whose  increased  density 
is  greatest  at  or  near  the  centre  of  the  wave,  diminishing  thence 
to  cipher  at  the  front  and  rear,  and  whose  particles  respectively 
have  the  velocities  due  to  the  respective  forces  by  which  they 
have  been  compi-essed;  and  they  have  illustrated  their  view  of  the 
nature  and  constitution  of  a  wave  by  a  diagram  constructed  sub- 
stantially as  follows;  where  AB  is  a  line  of  particles  over  which 

c  d 

a  wave  is  passing  from  A  to  B;  cc?  is  the  breadth  of  the  wave; 
and  an  ordinate  from  the  curve  above  to  any  point  in  cd  repre- 
sents the  compressing  force  and  velocity  of  the  i)article  at  that 
point. 

In  the  iifth  of  the  preceding  articles  a  sonorous  wave  is  defined 
as  consisting  of  contiguous  particles  of  air,  all  equally  com- 
pressed and  having  the  same  velocity.  Sucli  a  wave,  shown  by 
a  diagram  constructed  on  the  same  plan  as  before,  would  show 
the  wave  in  the  form  presented  on  the  line  CD;  with  a  hori- 
zontal line  above  and  precipitous  termini  in  front  and  rear. 


C____ I  I D 

According  to  the  view  of  a  wave  as  presented  on  AB,  the  qui- 
escent particles  of  air  as  they  are  successively  encountered  by 
the  advancing  wave  commence  their  motion  with  an  infinitesimal 
velocity  which  is  augmented  by  insensible  degrees,  reaching  its 
maximum  at  the  centre  of  the  wave;  but  a  wave  of  such  form  as 
is  shown  on  CD  requires  the  full  maximum  velocity  to  be  imparted 
to  the  quiescent  particles  instantaneously. 

To  this  last  view  of  a  wave  the  objection  is  made  that  a 
finite  velocity  cannot  be  impressed  upon  matter  instantaneously. 

In  making  answer  to  this  objection  it  is  proper  to  say  at  the 
outset  that  the  term  instantaneously,  as  used  in  the  foregoing 
articles,  does  not  import  an  absolute  negation  of  time.  It  is 
employed  in  its  more  common  sense,  to  indicate  a  time  shorter 
than  any  assignable  time.  Thus  understood  the  view  objected 
to  is  in  perfect  harmony  with  that  universal  law  of  dynamics, 

FT  X  MY 


60 

or,  The  product  of  force  and  time  is  as  the  product  of  mass  and 
velocity. 

For,  let  V  represent  the  absolute  velocity  imparted  to  the 
])articles  of  air  encountered  by  the  wave  ;  let  F  represent  the 
constant  force  required  to  overcome  the  inertia  of  those  particles 
in  being  put  in  motion  with  the  velocity  V  ;  and  let  M  be  the 
mass  of  air  put  in  motion  with  that  velocity  in  any  time,  T.  Then 
F  will  be  as  Y,  and  M  will  be  as  T,  and  consequently  if  we  make 
M  infinitesimal,  T  will  be  an  infinitesimal  of  the  same  order.  We 
have  then,  a  finite  velocity,  V,  imparted  by  a  finite  force,  F,  to 
an  infinitesimal  mass,  M,  in  an  infinitesimal  time,  T;  thus  sus- 
taining the  view  which  was  objected  to.  Tn  this  case  I  regard  the 
infinitesimal  mass  as  taking  the  velocity  V,  without  first  passing 
through  the  smaller  velocities  that  may  be  assigned  between  V 
and  cipher;  and  I  know  of  no  law  of  nature  which  precludes  the  as- 
sumption that  such  is  the  fact.  If,  however,  the  objector  thinks  that 
even  in  this  case  the  velocity  V,  is  acquired  by  gradual  acceleration 
and  can  conceive  how  this  gradual  process  can  be  carried  through 
in  an  infinitesimal  time  of  the  highest  order  and  within  the  limits 
of  a  space  equally  minute,  such  a  view  will  not  conflict  with  the 
views  maintained  in  the  fifth  article  nor  call  for  any  modification 
of  the  form  of  the  wave  as  presented  on  the  line  CD. 


Objection  III. 

It  has  been  objected  to  that  part  of  the  fifth  article  where  I 
have  estimated  the  extent  to  which  the  tem})erature  of  a  M'ave 
must  be  raised  by  evolved  heat,  in  order  that  its  velocity  may 
be  increased  from  916  to  1090  feet  per  second,  that  I  have 
assumed  that  an  elevation  of  the  temperature  of  the  wave  one 
degree  by  evolved  heat  will  have  no  more  etfect  to  increase  the 
velocity  of  the  wave  than  an  elevation  of  the  temperature  of  the 
medium  one  degree  ;  it  being  alleged  by  the  objector  that  one 
degree  of  heat,  confined  to,  and  acting  wholly  within  the  wave 
itself  must  have  a  vastly  greater  effect  to  increase  the  velocity 
of  the  wave  than  the  same  amount  of  heat  diffused  throughout 
the  medium. 

This  objection  seems  to  ignore  the  distinction  between  degree 
of  heat  and  quantity  of  heat  :  but  I  forbear  to  assume  that  the 
objector  has  overlooked  a  distinction  so  wide  and  so  obvious, 


61 

Other  things  being  equal,  the  velocity  of  a  wave  is  directly  as 
the  square  root  of  its  elastic  force  and  inversely  as  the  square 
root  of  the  specific  gravity  of  the  medium  in  which  it  is  propa- 
gated. An  elevation  of  temperature,  whether  of  the  wave  or  of 
the  medium,  has  no  effect  to  increase  the  velocity  of  the  wave, 
except  so  far  as  it  increases  the  ratio  of  the  elastic  force  of  the 
wave  to  the  specific  gravity  of  the  medium.  To  elevate  the  tem- 
perature of  the  vmve  increases  that  ratio  by  increasing  the  elastic 
force  of  the  wave.  To  elevate  the  temperature  of  the  medium, 
increases  that  ratio  by  decreasing  the  specific  gravity  of  the 
medium.  Hence  the  question  raised  by  the  objector  is  whether 
equal  increments  of  temperature  in  the  two  cases  result  in  equal 
increments  of  that  ratio.  I  maintain  that  the  effects  of  the 
elevation  of  temperature  are  precisely  the  same  in  both  cases  to 
increase  that  ratio,  and  to  increase  the  velocity  of  the  wave. 
For  an  elevation  of  the  temperature  of  the  wave  one  degree,  by 
increasing  the  elastic  force,  gives  to  the  air  of  the  wave  a  ten- 
dency to  expand  and  reduce  its  specific  gravity  in  the  same  ratio 
that  the  heat  had  increased  its  elastic  force.  An  elevation  of  the 
temperature  of  the  medium  one  degree  causes  it  to  expand  and 
reduce  its  specific  gravity  in  that  same  ratio.  The  effect  there- 
fore upon  the  velocity  of  the  wave  is  the  same  in  both  cases. 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 
BERKELEY 

Return  to  desk  from  which  borrowed. 
This  book  is  DUE  on  the  last  date  stamped  below. 


JUN     4  194J 


IrEC'D  ld 


LD  21-100m-9,'47(A57028l6)476 


